As the wood dries, it shrinks. You'll likely get big splits and cracks. The rule of thumb for a board (vs. a round) is a year of drying per inch of thickness. Wood dries faster through the end grain but also tends to develop splits and cracks as a result because of the uneven shrinkage (but has less tendency to warp in that orientation).
You can slow drying by sealing the end grain, but on a round, that leaves a lot of wood for water to migrate through, so drying could take years. A round two feet in diameter is pretty big and limits some of the tricks that can be used to speed up drying.
To do this in a "reasonable" timeframe, my inclination would be to not seal the end grain. Cut a bunch of rounds and let them dry for about a year until they stop losing weight. Assume you will get cracks and splits and just deal with that. Pick the best one to work with.
Cracks and splits aren't necessarily cause for rejection. A common way to deal with them is to fill the gaps with resin. Colored resin can turn it into a piece of art. Note, though, that the resin will limit the tools you can use for the engraving, at least on the resin portion.
Carving the logo
If you have access to a CNC shop, you can give them the round and a computer file of the logo (they will typically be able to create the computer file if all you have is artwork or a bit map image). They will be able to produce a perfect engraving, with very smooth and precise lines.
You can do this by hand; it will just require some time and being careful. A router could be used, at least for the larger lines, but it will be hard to see what you're doing for the small details. A rotary tool, like a Dremel, would make it easier to see what you're doing.
You could even use carving knives and gouges and do it the old-fashioned way. Unless you are an experienced carver, those tools are likely to give the result more of a handmade appearance, which might even be better for the look it sounds like you want to achieve. Knives and gouges won't be practical on resin, though, so if you have resin fill where you need to engrave, you could use a rotary tool in those spots.
There are a number of related terms you could investigate. Wood "engraving" is often reserved for fine, shallow cutting used to reproduce detailed pictures. The lines are cut into the wood (a "negative" image). "Woodcut" refers to a process where the background is cut away leaving the lines; it typically produces a cruder image. Patterns of thick, deep lines are usually referred to as just "carving".