beardoc 82, I forgot to mention a few details that you might want to take into consideration, which is why I decided to post again. Others may recall other options that could help you.

Flying to LHR, then connecting with BA to MUC requires the bus transfer from T3 to T5, then going through security at T5 and on to your gate. You mentioned that you are checking luggage, so you might take time into account. Once you arrive in MUC, it is then when you clear customs. I just did this past weekend, and it took 45 minutes @ MUC. They only had 2 lines open, EU and other passport. This was time consuming. I suggest that your transfer time at LHR upon arrival should be around 2hrs. I did it on 1h45m and had time to spare, but I only had carry on luggage. The same holds true upon return.

DFW-FRA-MUC hbf (train) is excellent, you probably can do the connection/transfer in 1.5h, maybe less. You arrive MUC HBF, which is the main station, in 3 hrs. Not knowing where you are staying in MUC, you will then transfer to a local train S or U Bahn to your station, this will be an extra 15-30 minutes. On the return is when things get complicated. Not knowing where you are staying in MUC, I will use MUC HBF as the main departure station, if you go MUC hbf (train)-FRA-DFW this is where you are in problem, as the FRA-DFW flight AA71 has a 10:30 am departure, meaning that you should be arriving FRA by 9am at the latest. The rail service MUC-FRA has a 1 train schedule that leave MUC HBF around 5:15am with a 9am arrival at FRA. On top of this you need to consider that you still need to take either the S or U bahn to MUC HBF. That might be a bit to much, almost 5 hours of train ride.

You might be better in doing DFW-FRA-rail to MUC in the departure, and then returning via air from MUC-LHR-DFW or MUC-FRA-DFW, the latter one with a connection through LH.