Hi Dave,
Yes I have done the same thing.
What is your stacking solution? ST and DSS can do it, but I like involving ASTAP or Siril too for file extractions as they are pretty easy to use and maintain 32-bit data.
The way I went about it was this -- use Siril or ASTAP to extract separate R, G, and B files from both the L-eXtreme and SII stacks. Name them so you know what they are. For the SII you only care about the R, the other extractions can be deleted. Call that R your SII file. Then do the same with the L-Extreme but keep all channels. The R you can name as your Ha.
At this point I would use ASTAP or DSS to register all four files - SII, Ha, and the G and B from L-eXtreme. ASTAP has a "cal & align only" option that will spit out aligned files. In DSS you would turn on save aligned intermediate files (and discard any resulting autosave).
The G and B from your L-eXtreme need to be combined into one. You can use ST composition for this and take advantage of how ST can bump it up for 2xG. Load the appropriate extractions into G and B channels, and pick a bicolor (GB)(GB) solution, mono. I think (?) interpolation should be off but remain unsure (and might not make a big difference anyway). Again if you want I believe you can also take advantage of one of the 2xG OSC solutions due to better SNR in the green. This mono file can get saved off as your OIII.
Now you can composite as S, H, and O. There will almost certainly be some cropping of the overlap errors required. Also, you might want/need to run separate Wipes on each channel, save the linear file, and then composite those. All depends on the data, but SII (due to relative weakness) and OIII (due to the bandpass) can sometimes need strong Wipe, whereas the Ha is clean and you wouldn't want to apply the same aggressiveness. But I've had it turn out good either way.
I think the last time I did this was in February. Hopefully I haven't forgotten anything!