You probably won't be able to get professional-looking results with DIY lamination.
Thermal Lamination
For thermal lamination, the thinnest common thickness is 1.5 mil (38 microns), which a number of companies offer. USI has 1.3 mil film, also. USI once sent me a sample 9"x12" pouch of 1.5 mil film. It kept curling and rolling up around the adhesive side; it was very hard to get it flat. Film that thin is probably practical only with a roll laminator.
If it's for a book/booklet cover created from a single folded sheet, thermal lamination would need to be done before the cover is folded. After lamination, the cover will be hard to fold. Also be aware that thermal lamination typically relies on a lip of lamination beyond the paper. If it is trimmed flush with the paper, it will often delaminate with handling, or the film will pull off the top layer of the paper.
Using thin-film pouches with a home laminator, it would be good to have a laminator with a temperature control. Units without a temperature control will likely be calibrated for 3 mil film. If you overheat it, the result will tend to be wavy rather than flat. It is also important to use the cardboard carrier (which is silicone-treated so oozing adhesive doesn't bond with it).
If you want thermal lamination, don't rule out having it done with a roll laminator. Check locally with places like schools or print shops that might do a volume of thin-film lamination. You may be able to get a small quantity done "inexpensively".
If you need to use a pouch laminator, it will be hard to make a one-piece cover. A more practical approach would be to laminate separate front and back covers using standard pouches.
Cold Lamination
Pretty much the same issues will apply to cold lamination
Adhesive Book Cover
You can buy adhesive book cover film by the roll, like this. It's fairly inexpensive. You cut off a piece the appropriate size, trim and score it, then peel off backing paper as you apply it. Here's a tutorial. This film is really intended as a protective cover. If you look at the pictures in the tutorial, you'll see that the results don't look all that professional. There's also a non-adhesive film sold in rolls for this purpose. It's a crystal clear protective cover, but it is typically taped in place to hold it (again, not a professional look). Libraries often prolong book life with covers like this.
Durable Plastic Cover
An alternate approach is to not try to laminate or wrap your cover, but use a durable plastic cover. These are commonly available as either a heavy-duty clear plastic cover that becomes an outer cover, or an opaque plastic cover with a clear window.
Prescoring
Lamination makes a sandwich of plastic film, adhesive, and paper (commercially laminated book covers are typically laminated only on the outside face), and that is thick. When you fold it, the outside of the fold needs to stretch and/or the inside of the fold needs to compress. The plastic film doesn't like to do either. Paper does it by breaking some of the bonds between the fibers. Whether you laminate the cover flat and then fold it, or apply the lamination to the finished book, the action of either opening or closing the book will break down the paper.
Laminating flat and then prescoring the sandwich spreads the paper damage over a larger, controlled area, and the paper will survive the normal amount of opening and closing. Prescoring the plastic and then adhering it to the finished book creates a plastic corner that needs to go somewhere when you open the book. That will either cause faster breakdown of the paper, or delamination (failure of the adhesive or separation of layers of paper fibers).
So prescoring is important for a professional-looking result, and prescoring the flat sandwich will produce the best result. This can be done with DIY tools, but it may take some experimentation and practice to get it right.
Hot vs. Cold Lamination
In order to get a professional-looking lamination, the film adhesive needs to bond with all of the nooks and crannies of the paper surface, which isn't smooth at a microscopic level. Otherwise, there will be trapped air that will look cloudy, or even form visible bubbles. Heat lamination melts the adhesive into a liquid and then pressure rollers force it into all the iregularities in the paper surface. When the adhesive cools, it's solid and doesn't flow.
Cold lamination adhesives don't change how they flow, or flow to the degree of liquified thermal adhesive in terms of surface irregularities. But they can be good enough if you apply pressure as the film is being applied, especially if the paper has a very smooth surface. Because the adhesive doesn't change how it flows, the film can "creep" on the surface, or delaminate at folds, over time if there are dimensional stresses.
To get good results with cold lamination, the pressure applied needs to be appropriate for the specific product. You need either the equipment designed to work with the product, or something else that mimmicks it. Trapped air must be squeezed out as the film is applied. That's normally done with a roll laminator, which applies the necessary pressure across the whole width as the film is being laid down.
You could sort of mimmick that action manually by rolling the film so the adhesive is on the outside. Then start at one edge of the cover and apply pressure on the film as you unroll it onto and across the cover. If you have a pouch laminator capable of being run cold, you could run the result through that, or go over it manually with a roller.
However, you need at least the pressure that the adhesive is designed for to force it into the surface iregularities. Also, once the film is down, whatever air has been trapped has nowhere to go. So you may have limited ability to improve the result with subsequent pressure. You may even find that there is temporary improvement, but over time the film looks cloudy due to trapped air and/or the film creeping back to its original configuration.
One other thought: you might want to investigate the longevity of cold lamination. Thermal lamination lasts "forever". Cold lamination adhesives might not have "archival" life, which you probably want for a book. It may well last for years, but not, say, a decade.
DIY Potential
Books are produced commercially with laminated covers, so it can obviously be done. That process entails commercial-grade equipment and supplies. On a mass-produced scale, it's even inexpensive. On a smaller scale, it isn't obscenely cost-prohibitive to do it with the proper equipment. You've already priced the rolls of film. Amazon has several roll laminators under $400 USD. If you amortize the equipment over thousands of laminations, the cost per copy isn't bad. If your requirement is for dozens or even hundreds of copies, it would be a lot cheaper to have it done commercially.
The question is whether you can replicate a professional-looking result without a roll laminator. How professional is sufficiently professional? Is the goal to share decent-looking copies with associates, or results that look commercially-produced for sale? If it's the latter, that's theoretically possible, but it would be very time consuming and probably entail a lot of experimentation and wastage.
If your total requirement is a few copies and you have the time, skill, and inclination to play with it, give it a try. The film manufacturers often have small trial amounts for the cost of shipping, or may send you a few feet for free. If your requirement is a lot of copies, consider how much your time is worth, and how much money you're willing to spend on it. You can put plastic on a cover easily enough. Getting it to look like a commercial result without the proper equipment is the problem. Roll laminators were invented to make this task practical.
