You can totally put them outside with the heat lamp...
You don't have to have a correct temperature brooder...you just need a spot in the brooder that is the correct temp...
So you hang the light and if the birds are cold they will get under it...if they are warm they will move away.
No need to check the temp, or anything of the sort...
That is why people end up habitually overheating their birds..they try to make sure their whole brooder is 90 degrees, which is so not necessary...worse for the babies in it, and slows down the feathering time.
Then again all my outdoor chicken enclosures have floors, or are off the ground, so the rain, snow whatever has no effect...
I guess keep them in until it dries, or put a piece of plywood under the tractor...and then put them in it..
THey won't get the benefit of being on the ground, but if the ground is all soggy they aren't missing anything.