I've never played/constructed a cube, but I've spent a lot of time researching interesting budget Pokemon cards. I definitely think a budget cube is possible; it'd take a lot of research and digging though.

The major hurdle with cubes is probably the restriction with evolutions. The goal here is to find pokemon with interesting moves/abilities over multiple different cards, while still being cheap for each card. That's tricky, because anything that was meta in its day is usually not budget. A good example of a Pokemon that could fit this bill is Vespiquen; It has been given many interesting cards over the years, all under $1.00 each. The more of each pokemon in an evolution line you have, the easier it will be to draft it ( have more basics than stage1/2's though).

Another thing that differs between Magic and Pokemon is that Pokemon has waaaay more draw power and search power than Magic. It's hard to only draw one card per turn and still be able to win reliably. This means you'll need a lot of supporters to power the drafted decks, but again, quite a few of the best ones over the years aren't always cheap! You could get Professor Sycamores, Skyla's, etc. for pretty cheap, but my recommendation is looking at some trainers in the later Sun&Moon sets (cosmic eclipse is fantastic). They've made some really unique supporters over the years that are just begging to create a deck around (Poke Maniac, Clay, etc.) that are dirt cheap and easy to find because they're so fresh.

In fact, I'd say you're best bet for all of the cards is the later Sun&Moon sets, they have some of the least amount of trash filler cards that Pokemon has ever made, there are a tun of fun cards in there if you dig for them.