Part of the Flat-Earth Expedition is to make a flat-earth map. Please help by donating and we can get underway sooner.
Since raising enough money for the proposed very ambitious expedition seems to be a problem, why not scale the first effort back and start by mapping, say, North America, using flat-earth principles. As you may know, most of the US states (other than the original 13 colonies and Texas) are surveyed into mile-square blocks called Sections. Because the original surveys were carried out using a spheroidal-earth model, however, once the surveys progressed some distance from their Baseline and Reference Meridian, the sections became distorted. Since surveying a flat earth into a regular one-mile grid should be really easy, why not do your own section-line resurvey and you can point out to professional land surveyors where their mistakes are. Much of the Great Plains in the US is crisscrossed by a grid of section line roads that are open to the public, so access to vast areas shouldn't be a problem, and if you start some place like, say, Iowa, logistics should be a snap since the terrain is gentle and small towns and cities every few miles are available to replenish supplies. Operating out of small mom & pop motels or even camping could be done on the cheap and be much easier than whatever you'd have to do in remote regions with a very hostile climate. This seems an ideal place to hone your techniques.
Highly accurate surveyor's transits, tapes, and chains that are no longer used professionally should be available for a song. The pros now almost always use electronic equipment that you probably wouldn't trust, so simple manually-operated instruments with engraved scales and bubble levels, along with steel tapes or chains, in addition to being cheap, would seem to fit your style.
If the value of your ideas can be clearly demonstrated from a non-trivial but still manageable project like correctly mapping Iowa, you'll have no trouble raising funds for
much grander projects. After all, if you're successful, you'll want to do this anyway, and if you find out that a mapping project on this scale is too difficult, then whatever do you think will happen in truly wild places?
Starting modest has obvious advantages although it certainly isn't as sexy sounding as an all-out assault on Antarctica. Think of it in practical terms, though. And, remember, the earth wasn't mapped in a day!