This is one of those questions with some different permutations of what you could do and what the compromises might be.
FC-M600-2 appears to be a 4-bolt 104/64 symmetrical crank, which is not what you need. Also, it generally will not work to take a ring for a crank that's made to mount on the outer spider position, where the bolt counterbore is on the outside of the ring, and adapt it to one of the newer style doubles like FC-M4100-2, where the large ring mounts on the inside of the spider. In some cases you might be able to make that work just by machining or grinding your own counterbores, but beyond that there's the question of will the ring-to-ring spacing still be right even though you're using a spider contact surface that wasn't intended. In some cases that surface might not even be flat. I haven't experimented much with that and don't have all the answers to it, but suspect it won't go well because the design of the two kinds of cranks and rings are quite different.
The bolt pattern for the 38t ring (WP-Y1WD98030) from FC-M6000-2 is the same as what you need. That ring isn't listed as compatible by Shimano with your crank and has fancy 3d contouring around the mounting holes, so it may interfere with the spider, but that may be a problem that could be fixed with a dremel or file if it doesn't just go on. I suspect that's likely true. If you did that and it goes on, you're left with two other potential issues:
- The shift aids on it are intended for a 28, not a 26, so the shifting will probably be laggy. I don't think that's a problem you can get around with any existing 10-speed 38t ring for Shimano 96mm asymmetrical double cranks.
- Your front derailleur capacity is 10t (doesn't want more than a 10-tooth jump between rings), which means it won't be usable at some point in the small/small combinations. That is a problem you can solve by getting an older 10-speed double mountain front derailleur. A lot of the earlier Shimano 2x10 mountain ones have capacity of 12 or 14, say from the 2010-2012 era. FD-M785 is an example. You can look up this data in the archives on si.shimano.com, and then go on an ebay goose chase to find one. I couldn't figure out a new example of a specimen like this that's actually available - there's probably technically one that's still made as a repair part, but I couldn't discern what it is, and there may not even be one.
If keeping the range the same and just moving it higher is acceptable, then getting the 28/38 set for FC-M6000-2 and doing the above mods to the mounting area if needed is likely to be a simpler answer, and should work without issue with your current FD. Make sure you've got the frame clearance to do it; many more recent bikes play every trick possible to get their tire clearance, and will be unforgiving about this. The same is true for going to a 38 by itself.
Note that there's usually a tooth offset difference between Shimano 10 and 11 chainrings. If you find an 11-speed ring that appears to do what you need, that's another thing you would potentially need to deal with, although it might be a problem that could be solved with micro shims under the small ring.