Unfortunately, I don't think there's any way to do it in LOTW. You only enter one mode, and the two have to match up.
I don't think there's any reason why the contact wouldn't count for most awards. For example, for a basic WAS, there isn't any restriction in the rules for mode that I'm aware of. But there's no way to do it with LOTW.
Interestingly, that reminds me that I got a QSL card from one guy that didn't have the mode listed. It didn't even have a spot for on the card for the mode. IIRC, it was one of those slightly difficult close-in states on 15 meters, so when I get around to finishing up my 5BWAS, I'll have to see if it's valid, if I don't work another one in that state before then.
I also recall working one or two cross-mode contacts as a novice. Kindly Canadians occasionally came down to the 15 meter novice band and answered in phone. That was a big thrill at the time.
On the log you log your mode under "mode" and on the notes side log his mode. Fcc cares more about what your radiating and not so much what your receiving -hi