However, price affects how many people choose to enter (or exit) the mining, thus changing the difficulty.
Right, but if the price is too low to justify starting new mining (meaning buying equipment to mine) and people want to transact in btc, then they will have to buy it. This will put the price up. This is why I think these three, price to enter mining, price of a btc, and difficulty, are all related by an equilibrium.
Well, if someone is looking to spend in BTC and they choose to acquire it via mining, then that one less block (or part of the block) in the supply side of the market for currency exchange, which effectively makes the price go up (or not go down as much) all else equal. If they instead let someone else mine it and buy that 50BTC, theoretically the price should go up about the same (or not go down about the same), assuming everyone's preferences for BTC and USD are the same in both situations. Again, it does not affect price.
Price is the intersection of quantity demanded and quantity supplied. In your situation that quantity does not change. If price to mine jumped up to 100x the price it would not change the quantity demanded or supplied for BTC, just shifts where people spend their money to acquire it (assuming difficulty adjusts fast enough, which is the only lagging factor than can affect price because it means more (or less) than 6 blocks an hour on average are being created)