Playing C Ionian, F Lydian and G Mixolydian, exactly the same notes are being used - C D E F G A and B. As you say, you are aware that this is the case. For guitarists, it may not be that obvious, as the pattern learnt for Mixolydian/Lydian/Ionian may not reveal the same notes - they are the same, but with a different pattern comes a different mindset. It wouldn't be the first time I've encountered this thinking!
However, the question. On an F bar (in key C), the F scale notes can be made to work; the only difference is Bb instead of B. It's going to depend where in the phrase it gets played. The same with G scale notes; F# in place of F, which actually gives the mode notes of C Lydian.
So, instead of thinking only modally, or only key scalarly (made that up!), use all the available notes. Now, the choice is from C D E F F# G A Bb B. In fact, a Bb over a G chord will work in that it makes a bluesy feel - what's wrong with that?
The more important factor is to use chord tones on the main beats, as, for example, when on G, a G B or D note on the first and/or third beat of a 4/4 bar will make it sound like it fits well.
When all that is coming more naturally, and there's less thinking involved, expand each chord. Like, on an F bar, use b7 as well, (no point mentioning the 9 or 11 or 13 as they're already available from the F scale!).
I've kept it straightforward here, but the ideas could (and jolly well are!) expanded every time someone extemporises in most styles - obviously jazz comes to mind, but I'm sure Bach, Listz et al used the same principles, too.
Bottom line is the usual 'if it sounds good, it probably is' - all of which can and will be explained in theory, but quite often players don't allow themselves to let theory lead - there are many good players out there who are ignorant of the theory, and it hasn't slowed them down.