No, this phrase isn't cognate with Standard Japanese あした.
したっけ literally means what in Standard Japanese そうしたら. The demonstrative そう is omitted because the whole context before is considered to stand in place of it (colloquial omission of this そう is also common in Tokyo). The っけ part shares the same origin with Standard っけ ("(what) again?"), that is Classical indirect past けり, but has diverged from it to mean "(after it) then".
Thus, Hokkaido-ites say "Now then!" to mean "See you!", but this is exactly parallel to Standard Japanese too, where さようなら literally means さよう ("so") + なら ("if be — then") in older way of speaking.
したっけ is seemingly used in the same meaning in Ibaraki dialect, too. Hokkaido dialect is a mish-mash of various Honshu dialects, so it's possible that this part of grammar was exported from Northern Kanto.