Since you are on a DONE4 you will have a J boarding pass on BA, allowing you to use the FCC FastTrack -- which takes 5 minutes instead of 1 hour+ in the regular line. There is no problem with 85 min connection T1 to T4 using FastTrack -- the problem is for those who aren't FastTrack eligible (and BA polices this ardently). BA Gold doesn't get FastTrack at the FCC, hence all the complaints on the BA forum. As for baggage making it, I've never had a bag miss a short connection (it will be tagged as SHORT in FRA).

Finally, the law is murky about whether overnight inter-airport transit qualifies for exemption from the APD and most airlines collect it, so that would add GBP 80 to the cost if you use LCY. It will be collected at LHR upon check-in if not already on your ticket.