When rejecting DIY solutions, you are limiting your choice a lot. Methylcellulose (MC)-glued double and triple tissue and well as self-made tissue foil are very popular and relatively inexpensive materials for folding exactly the kind of models you describe.
The situation is not hopeless, however. There are ready-made papers which may suit your needs, but you will usually have to pay dearly for the benefit of not having to prepare paper yourself.
- For practice, Kraft paper is often very good. It is widely available, strong, and you can get it in a range of thicknesses, including thin (e.g. 40 gsm). It does not look too good, but its folding properties are quite nice and it is inexpensive. You can usually get some variant in a local art supply store and save on shipping fees. This kind of paper often comes in rolls in which case you can get large formats, even above 1 m, easily (you have to cut the square yourself in such case, though).
- Nicolas Terry Tissue Foil is a ready-to-use replacement for self-made tissue foil and seems to be thinner than what you can make yourself.
- Various mulberry papers can be good for this sort of models. They are available online and sometimes in art supply stores (depending on where you live).
- A broad range of Washi papers are suitable for complex models. Be careful, however as Washi can mean a lot of different things, from very thin to rather thick. Be sure to check the weight / thickness of the paper.
- A number of hand made papers may be suitable but since they are all unique, and often do not even have an official weight rating, it may be hard to find the right one without touching it or having a qualified seller who knows origami choose one for you.
- And of course at the high end there's some Origamido available online. That's what Robert Lang often uses for his best models, though it comes with a price tag of €13 per sheet.
These were some specific paper types which may (or may not) fit your model. Note that each particular model may have different requirements on thickness and stiffness of the paper. For example, Shuki Kato's elephant can certainly be considered a complex model (196 steps) but it can be, and very often is, folded from Elephant Hide paper (110 gsm) which is an absolute no go for most insect models.
So, think about a particular model. What you can do is search flickr or other places where lots of origami models are displayed for your model. People will often add the paper type and weight to the model's description and this can help you choose paper for your own folding. When you buy online at a specialist origami store, you can also ask the shop about particular paper's suitability for certain kinds of models. No one can guarantee you a good fold (it depends very much on your own work), but they can give you hints which paper may be good and which you should avoid.
Ilan Garibi has a series of paper reviews on OrigamiUSA where he has reviewed a number of commercially available papers. I have found these reviews to be very helpful when I started to get interested in origami papers myself.
Finally, here are some properties of the paper you should look for. Generally, you want your paper to be thin, yet strong and crisp. In more detail, this means paying attention to:
- Paper weight, usually expressed in gsm (grams per square meter). These values are only approximate ratings but they give you an idea of how thick the paper is. 80 gsm is the weight of standard copy paper and is too much for many complex models, for which you should look for 40-60 gsm paper. Some very fine papers used for insects go as low as 20 gsm. Elephant Hide, a thick and stiff paper, is rated at 110 gsm. Papers around 160-300 gsm are suitable for wet folding but hardly foldable dry.
- Fibre types: hand-made papers have long fibres which is usually a good thing. Longer fibres make the paper stronger and make it behave better when you need to wet fold some detail.
- Crispness: it should be possible to make a sharp, well-defined crease at exactly the place you want it and paper should hold the crease well. This is especially important for complex folds like sinks where you precrease a line and then collapse along it. Soft paper has its benefits too, but too soft may make folding difficult.
- Aging: one of the reasons I started using better papers was I noticed I had to throw away a number of old models which had faded and become very ugly. If you invest a lot of time into folding a complex model, you want it to stay looking fresh for a long time. Chlorine-free and acid-free papers are expected to last longer than those with the extra additives. Hand made papers often avoid these chemicals and age quite well.
The first paper review by Ilan Garibi describes these and many more properties which affect how origami paper behaves when you fold it and which make it suitable or not for particular model types.