$600 tops for everything.
The recordings will have no particular use for now. But my hopes are in XX number of years I'll have a decent library of ambiance, foley and general sound fx to offer as a sound effects library.
Here's maybe a far out idea for you.. and is what I would do, but I'm biased to this direction. Two variants:
1) Use your existing DR680 and buy 4 or 5 miniature omnis with decent sensitivity and self-noise. Buy phantom power adapters to step the 680's 48V down to PIP for the miniature omnis. Alternately you could use Plug In Power battery boxes or preamps to power the mics. You can do all that within your budget. Attach the omnis to each side of a 3-dimentional baffle like a box or cylinder, with them facing all horizontal directions. You can put the recorder and battery inside the box. That arrangement would be no more bulky than the cage with mics you linked to.
I regularly use 4 or 5 miniature omnis that way, powered with Niant PFA adapters (Niant = Jon Stoppable) into the DR-680.
2) Use four of the same miniature omnis with decent sensitivity and self-noise as above. Buy a Tascam DR2d (recent recurring deals has is new for ~$100), which can record 4 channels as two stereo pairs simultaneously, but provides no phantom power. Power the miniature omnis with two battery boxes or small preamps. I use Church Audio CA-UGLY preamps (build by another small vendor who is a member here) which are very small and provide gain. The entire rig will fit in a small shaving kit or in pants pockets. I do this with DPA omnis, which are outside your price range, but you could do all this with less costly mics. Jon may build a suitable low cost diffuse field type omni which would allow you to buy the recorder, mic preamps and mics, all within budget.
Then record your ambient sounds and spaces in 4 or 5 channel surround. You can always discard the extra channels and have two channel stereo, but the material you mention will
really benefit from surround playback immersion and having it in 4 or 5 channel format will give you more options for doing interesting things with it, even if you can't listen to it that way now. I record music that way so I obviously think it’s a valuable and highly rewarding way to record, yet I’m sort of the odd man out around here on that account. Yet for what you are doing, recording in surround is an even more well suited match. Consider it.
I wish I had such a setup when I was in Asia.