Paralleling subfeed lug blocks is an option...
While the THLK2200 indeed is only supported by PowerMark Gold loadcenters, there is nothing that prohibits one from putting subfeed lug blocks (as opposed to actual breakers) in parallel, provided they support attaching large enough wires (1/0 or larger), and the attached wiring obeys the NEC rules for parallel conductors, namely that the wires must all be the same length, use the same kind of wire, and be derated appropriately if they're in the same conduit. This lets us use a pair of THLK2125s to get 200A total out of this panel, despite the lack of a single 200A subfeed lug block that'll do the job.
Furthermore, one can also use subfeed lug blocks to backfeed a panel provided they are held down with the appropriate retainer kits, just as one would do with a backfed main or generator breaker. This allows us to come up with a way to handle the length-matching issue without undue difficulty, given that your panel does not have any global restrictions on stab ampacity given on its label.
Of course, you'll need neutral lugs to match, and for this, we'll use a 286A8894G1 add-on neutral lug kit, mounted to the left-hand inboard neutral lug position at the top (the "tab" there), along with the existing 2/0 lug at the right-hand inboard position. The two subfeed lug blocks, then displace the existing breakers at the top of the existing panel, going across (these breakers will have to move no matter what, since your panel only permits T(H)QDL breakers to be fitted in the top four full-width spaces).
As to the feeder itself...
For the feeder, we can easily get 6 1/0 Al (or even 2/0 Al if you want some extra safety margin in case something comes undone) XHHW-2 wires in a 2" rigid nipple fitted between the two loadcenters, coming out the bottom of one side of the existing panel and going into the bottom of the adjacent side of the new panel. You'll want to order a single length of XHHW-2 Al wire for this and cut and tape it yourself (black, red, white is fine) so that you can ensure that you have equal lengths of wire and that everything is correctly phase taped so you don't accidentally short something out by way of mixing your wires up. The conduit nipple, then, serves as the grounding path between the two enclosures; you'll need to make sure that the locknuts are made up tightly for this, and that any paint on the insides of the boxes around the locknut contact areas is removed.
...and the choice of loadcenter
We already know that we'll need a 200A or 225A NEMA 3R (outdoor/weatherproof), main lug loadcenter for this new panel. The only question left, then, is a matter of spaces, and for that you'll want to go as big as you can get. If you're sticking with GE, this means using a TLM4020RCU or perhaps a TLM4225RCU; however, this limits you to 42 spaces, with no way to migrate breakers from the old loadcenter to the new since neither of those panels can accept half-width (double-stuff, THQP) breakers.
With these panels, you'll want to leave them configured as main lug, but leave the normal main lugs unused. Instead, you'll want to fit another pair of THLK2125s in the topmost set of spaces, going across, held down by a THQLRK4 backfeed retainer kit (one retainer kit holds down both subfeed lug blocks).
The incoming feeder neutrals, then, land on both neutral lugs in the new panel, one per lug, and a TGK42 ground bar will need to be fitted to the new panel as it's a subpanel. You'll also need to be careful when wiring up the new panel to make sure that both wires connected to the A leg in the first panel connect to the A leg in the second panel, and likewise with the B leg, lest you create a 240V short that the main breaker in your existing panel will tell you about with no shortage of urgency when you try to turn the power back on!
If you don't want to go through this hassle, though...
If you find the above process to be too much of a hassle, though, there is a way to get a 200A breaker for your panel, it just takes a bit of cleverness and understanding of corporate parentage. You see, GE has a subsidiary known as Midwest Electric that makes RV pedestals, metering equipment, and such; what's not as commonly known is that this equipment uses GE breakers, despite the fact that they have their own breaker part numbers. So, as far as I can tell, the Midwest CB2200B should be equivalent to a TQDL21200, and somewhat more available as well (KSCdirect has the former for about $300 at the time of this writing).
With this, we can use the same 2" nipple to the new panel, but a somewhat more conventional feeder consisting of two 250kcmil Al hots, run from the lugs on the new feeder breaker to the main lugs on the new panel and a pair of 1/0 or 2/0 Al neutrals, run from the top set of neutral lugs (with a 286A8894G1 fitted to the inner left-hand neutral lug mounting position, again) on the existing panel to the neutral lugs on the new panel. In this case, phase taping isn't critical, but it's still important to make the paralleled neutrals the same length! (Your panel apparently can't accept the larger neutral add-a-lugs for some reason, based on its labeling, and I can't find any instructions that say otherwise.)