I too have had trouble muting strings, and one way to kind of bypass the issue (though you really should develop muting technique) is to find another inversion that doubles a note instead of having to mute one. I this case, a few inversions work here, like:
5-5-5-3 (left side is G-string), then 5-5-5-5
or the less comfortable
5-5-1-0, then 7-5-1-0
Both of these play the exact same notes as the muting passage on the paper, they just play a note twice. Additionally, one can do the strumming easily with these inversions as well. If you really can't mute, the first one's your best bet. If you want to learn how to mute this particular chord, if you know how to play an Fmaj7 chord like this (2-4-1-3), then you can just not press down on the C-string when you play it. You could also try a triple-stop, where you just pluck all three notes at once, but that loses the sound of the downstrum. Another thing I've seen some players do is just leaving the G-string opne at almost all times, and on this chord you could play some 3-note F-chord high on the neck and leave the G-string open, which would sound different but not too bad.
TL;DR: Options one has are:
Learn to mute (ideal)
Inversions that avoid string muting (optimal alternative)
Triple Stop (Not a strum, so loses downstroke feel)
Leave the G-string open (sounds different, but it could sound really cool too)
Remove one of the strings from your instrument (If you want to get really radical)(Please do not do this)