Probably hard to do, not impossible.
Sad to say, the first revision of my RV7A, with basic (and I mean basic) panel and no paint was $70K or so in receipts.
Add what I have now, (updated panel and new paint) and it's north of 90.
So, to answer your question, yes, you probably can build an RV and stay in the 30s. It won't be easy.
Find a 3 4 or 6 that is partially completed, maybe even scrounging parts from many sellers at rock bottom prices.
Find a used, mid time engine with accessories (or even high time). Better yet, find a wreck with engine, accessories and panel in it (wentworth.com)
Panel, my original panel, was kinda high, because I chose an SL30 and a King Transponder. Even without an efis of any kind it was about $9K total.
If you scrounge used instruments (or purchase a wreck with a panel/engine in it), you might be able to get there.
It will be tough.
Just as an after thought, I have passed up more than one RV3 that was priced in the high 20s and a really, really nice RV4 for $41k. Heck I even passed on an RV6 at $39K, so there are bargains to be had.
Unless you really want to build, it probably would be cheaper to buy a flying airplane used. (as if anything about airplanes is cheap).