(It's a stretch, but worth a try) (I'm assuming here that you have already confirmed that FRA-HKG no longer has availability for two I seats that you can just book)

Firstly, the reason it would have given you married segments in the first place is usually because booking them as married segments give favourable availability - for example, FRA-HKG alone might not have had availability, whereas LHR-FRA-HKG might have shown to be available.

Anyway, in some cases, the following may work, although I'm really not sure whether it would in case of reward bookings (and inter-airline, at that)

When an agent opens your PNR and cancels a segment, in order to finalize the changes the agent must do an EOT (End of Transaction) command at the end. This then sets all the changes in concrete and releases the seats permanently. Before you do the EOT, you can always just press Ignore, and all the changes made will be ignored (and you retain the same bookings you had before).

In some cases, if the agent cancels a segment in your PNR, and before the agent does EOT, the seats will be released and show as available on the availability display, but only on the terminal that your agent is using - in other words, you press Cancel LHR-FRA-HKG, then it may then release those seats straight back into two I seats from FRA-HKG, but remain protected on your agent's terminal until the agent does EOT.

In short, it's like a semi-cancel - the point is that the DC agent might be able to cancel the LHR-FRA-HKG, then (without doing EOT) re-load the availability screen, and if it then shows availability for two I seats from BHX-FRA-HKG, s/he can request those seats straight away. (Then do the EOT, obviously).

I'm sorry if it sounds overly complicated - it's basically a small technicality on airline CRS, and might be entirely unapplicable to your case (especially because other airlines' reward seat availability tend to be not as well linked as revenue seats). But maybe you can call them to ask whether they 1. understand what you're talking about, and 2. whether the suggested procedure actually works.

PS. I just remembered - BD's India call centres. Umm...

Otherwise, it looks like a very, very early coach for you from BHX to LHR! Good luck!