I gave Fun 4 a go briefly in Lemmix for what I think an optimal TAS solution might look like when translated back into Genesis. First, don't look at the game timer in Lemmix because DOS's goes down slightly slower than Genesis's, so they simply aren't comparable that way. Here are the things I tried to improve on:
1) I try to make the 2nd lemming out overtake the miner as soon as possible, assuming that the 2nd lemming out will be the last to exit and therefore should be optimized. The goal is to eliminate the amount of wasted frames due to any jumping and falling that would otherwise occur for that lemming to get up the mining tunnel's end. In Lemmix, it looks like you need the RR to be at least 40 before the 1st lemming is out, to completely eliminate any jumping and falling there (walkers can walk up a step of 2 pixels in the same frame of moving forward 1 pixel, so upward steps of 2 pixels or smaller are okay). I don't know if this RR increase is feasible in Genesis especially for TAS. You may have to settle for a value lower than 40, resulting in less reduction (2 frames I think will be wasted in jumping up a 4-pixel step, if you can't do RR 40).
2) The RR is then reduced to 29 before the 2nd lemming is out. This is the minimum amount of reduction from 40 that still ensures the 3rd lemming out cannot also overtake the miner. 40 -> 29 I'm guessing should be achievable, and if you can't even reach 40 in step 1 in the first place, then you have even less problem here, as you won't need to lower the RR as much.
3) As you can see, the 3rd and 4th lemmings out will turn around in the mining tunnel, then the 5th lemming out onwards will not need to turn around anymore as the mining is completed by then. You need to start raising the RR to maximum as soon as the 4th lemming is out (and maybe even a little earlier, as long as you can ensure the 5th lemming out won't turn around), to ensure the last lemming out will not be slower than the 2nd lemming out (the one that overtook the miner), who will then be the last to exit as predicted. I purposely didn't pause to do this final RR increase, to get a sense of feasibility. Obviously may work out differently on Genesis depending on how fast the RR can go up.
One thing I haven't optimized yet is the idea of having the 1st lemming out start mining a little further to the right. What this can achieve is to allow the 2nd lemming out to turn around sooner relative to the progress of mining, which can potentially reduce the amount of wasted time falling into the miner tunnel when it is walking left (because the mining tunnel wouldn't have reached as far downwards). Of course, doing so would also increase the amount of walking of the 3rd and 4th lemmings, and you don't want them to end up behind the 2nd lemming out. So there's likely something to tweak and test there to squeeze out yet a few more frames.