Is there a way to determine how much of a "fighting chance" I had?
A fat chance/A slim chance, whatever you wish to call it.

Even if you could get the last railroad AND the last utility AND get all hotels on your set for $0, you will still lose, unless a miracle happens, i.e. you never land on his hotels or on the set until his hotels are down.

There is a 7.54% chance (2 decimal places) you will land on his hotels (and pay at least $750) = equivalent to you losing $56.55 (2 decimal places) every turn (assuming you will land on his hotels at EXACTLY 7.54%)

There is a 4.29% chance he will land on your hotels (250 for Medit, 450 for Baltic) ($15.06/turn)

If you have the 4 railroads, 11.38% chance (gives $200) ($22.76/turn)

2 utilities: Average rent 7*10=$70, 5.41% chance = ($3.79/turn)

Total "gain": $41.61/turn

Total "loss": $56.55/turn

So this means that you may lose.

Note again that this is the most general case and I am only working with probabilities, that is, if the dice never or always leads to a space, this answer might be wrong.

Credits http://www.tkcs-collins.com/truman/monopoly/monopoly.shtml for all the probabilities