Engineer here. I dont care to do the math, but the 49F lake should work within a day or so (and with a minimum of effort expended). If you can hang it from a dock or something, that would go quicker than resting it on the lake bed. Constant contact with the water means you have a good mechanism for heat transfer, and a lake is a large thermal mass that wont change temperature due to the cold keg. The biggest advantage of using the lake is that your beer is never going to get warmer than the lake. You have a couple days to play with, so why push it with a hot bathtub? It would obviously be a good idea to hit the ball lock posts with some rubbing alcohol before proceeding with the next step.
Side note: When you thaw a turkey in the fridge, the air that's in contact with the turkey is an excellent thermal insulator, which is why it takes the turkey so long to thaw. If you put the same turkey in a sink full of water, it'll be thawed in an afternoon.
Most of the gas also came out of solution when it froze, so you're probably going to have to recarbonate. You can do that in an hour if you use enough elbow grease. Cool the keg back to serving temperature, then pressurize the cold keg, shake it up, and repeat. It'll take a bit. Remember that 1 volume of CO2 means that you have 1 gallon of CO2 (at atmospheric pressure, ~15 psi) dissolved into 1 gallon of beer. So 2.5 volumes means you're injecting 12-13 gallons of CO2 into the keg to get it carbonated. A bit of extra headspace helps reduce the number of times you need to charge it with more CO2. You can use higher pressure to speed the process too. Just keep testing it as you go so you dont overcarbonate.