I'm not 100% sure about how to play the 7th chord on that keyboard, but when I just tried it on my PSR-E433, if you want F#7 with the easy-play chords, you play the F# and then F just to the left of the F#, not the E. It looks like, regardless of whether you are playing a white-key chord or a black-key chord, if you want the minor of that chord, you play the root note and the black key closest to the left of the root note you are playing, if you want a 7th chord, you play the root note and the white key closest to the left of the root note you are playing, and if you want a minor-7th chord, you play the root note and both the black and white keys that are closest to the left of the root note you are playing.
This can result in some significant variations, depending on the chords you are playing. For example, to play a G minor chord, you play the G, and then the Gb (or F#) directly adjacent to the left of the G -- a "distance" of only a single semi-tone. But to play an F# minor chord, since you have to play the black key closest to the left of the root note (along with the root note), in this case, that is the D# (or Eb) a whole minor third to the left, which you of course also play along with the root note of F#.
Now, for that F6 chord? Fugghettaboutit! These simplified easy-play chord systems are not sophisticated enough to handle the numerous varieties of enhanced and jazz-type chords, and if they were, this would require enough keys to be pressed and enough complicated rules that you might as well learn how to play the actual chord! F-A-C-D, in this case, by the way.
With that said, a semi-workaround with a sixth chord would be to play the minor-7th of the chord a minor 3rd below the root of the 6th chord that you want, as that chord would contain the same notes. So, with our F6 chord consisting of the notes F-A-C-D, you could play the D minor7 chord of D-A-C-F -- same notes, just a different inversion. So, you'd get these notes with the easy-play system by playing the D, and then the black and white keys immediately closest to the left of the D, which would be the Db and the C. This will get you the D minor7, containing the same notes as the F6, but of course, it will also be playing a bass line based on "D", instead of "F", which is why I said this is a "semi-workaround."
Finally, don't worry about "bothering the group." That's what we're here for!