Which random money schemes do you think would be best at this?

Bitcoin, interestingly enough. Bitcoin has multiple competing blockchain lotteries which are often provably fair, and which generally take <2% in fees, while offering high-leverage bets; for example, SatoshiDice famously paid out 1920btc on a 0.03 wager.

These are using cryptographic hashes, which are unpredictable but deterministic, so they're not immediately applicable to MWI. What you could do for MWI purposes is use a quantum RNG to choose at what point in a day (or month) you bet at; since each block gives you a different chance at winning, quantumly choosing between enough blocks can essentially guarantee a win for at least one branch.

So, assuming you had the necessary bitcoins on hand, this is actually a very easy way to leverage up (far easier than some suggestions I've seen like trading foreign exchange options). Run the RNG, send it whenever, and within 10-15 minutes you'll have won a large fortune or lost a small one for a fee of ~1.8% of your bet.

I was actually thinking about this as a strategy for myself. I have a decent number of bitcoins (I began accumulating what I could back in May), but I don't have a life-changing number of bitcoins. On the other hand, if I bet at, say, 10:1 odds on SatoshiDice, that combined with future bitcoin price increases would be life-changing. I could fire up a quantum RNG, ask it when to bet over the next week, bet, say 25btc, and in one of 10 universes, walk away with ~248btc and cash out for $200k.

Is this a great idea or a terrible idea? I really don't know.