That probably wouldn't be a bad idea, considering Fleeting can easily be used to make tiles that can be played every single turn for north of 20 gold, 20 damage, and 20 healing each. As you run down the rest of the deck, unless the AI places blockers that make it impossible to play your combo tiles, it just becomes you being immortal and at the money cap waiting to do enough damage to beat the encounter. The only hiccup with making it more expensive is that makes it harder to encourage the kind of problem solving about each piece that leads to realizing just how strong Fleeting is, but you do at least start with 1 arrow in the default deck.
Making Fleeting and Persistent cancel each-other out may also be an avenue to explore, a bit like KSmarts suggested for Exile in another thread. Persistent tiles seem eager to throw large bonuses for being played, presumably because they're balanced around being applied once the entire fight. Up to you in the end, though. Balance is nice, but a few busted combos for clever players can be that spice that makes a run really stand out.