Which wood Grado

In my opinion....

You're better off getting an Otello Soundsmith cart.  Much better choice and it fits your budget.  And when you like the way it performs, it only costs $100 or so to rebuild to new.  And if you want "more" during the rebuild, maybe Soundsmith just might provide a better cantilever and / or stylus for a small increase in rebuild price.  If you ask nice.

The Grado Sonata V2 reference or statement. The difference between them is the output. Both would be a good match with the tonearm on the 1200GR. BTW, The Needle Doctor has a return policy if you don't like them!
https://www.needledoctor.com/analog/analog-components/Phono-Cartridges/brand/Grado-Labs

Can I get that with the $600 or $700 budget in the Wood line and which one would be the best. MM or MC.

All Grado cartridges are MI, not an MC:

"Although we were granted our patents for the stereo moving coil cartridge in 1959 we ceased manufacturing moving coil cartridges a few years later. We discovered that the moving coil design has an inherent issue that causes a high frequency emphasis which cannot be corrected through any type of engineering. This encouraged Grado Labs into patenting a platform of moving iron phono cartridges. The Moving iron design eliminated all of the known issues of the moving coils."

My favorite Grado is Signature XTZ model, made by Joe Grado in the 80s. Within your budget you’d better look for some top vintage MM or MI for your tonearm. Victor X-1II or X-1IIe are amazing just to name a few.

If you looking for MC then Fidelity-Research PMC-3 would be the best with heavy headshell and sub weight for your tonearm. Very nice cartridge, pretty rare. You can find more in Fidelity-Research thread.

Another MC with relatively high compliance would be Dynavector DV23RS MR (with Ruby cantilever and Micro Ridge tip). You can find more in Dynavector thread.

P.S. If you want ONLY brand new cartridge then you can buy Garrott P77i or higher model and report back how do you like them, some of us are happy with old version. They are all MM. Find more: https://garrottbrothers.com/ Replacement styli are available everyday and it’s much better than sending cart to re-tip (in case with MC).

My advice for your 1200GR:
Remove rubber mat and add Micro Seiki CU-180 gunmental mat or SAEC SS-300 alloy mat.