From the point of view of composition, you may want to initiate your verse with the I, and then see if you might use a V/V (D, F#, A) in whichever inversion to get the leading tone back to the G in the bass line. A V7/iii or VII7/iii could also give you that F# leading tone into your inverted I6/4 in the bass, and allow you to suspend the C tonic (perhaps in octaves) across another part or two. This is essentially just a tritone resolution to the key of G, which may fool the ear and make the inversion feel more appropriate.

Perhaps use the keyboard to play the C in octaves (which are inherently stable) when you arrive back at the one to offset some of the inherent instability of the inversion.

Be interested to hear what you come up with.

From the point of view of composition, you may want to initiate your verse with the I, and then see if you might use a V/V (D, F#, A) in whichever inversion to get the leading tone back to the G in the bass line. A V7/iii or VII7/iii could also give you that F# leading tone into your inverted I6/4 in the bass, and allow you to suspend the C tonic (perhaps in octaves) across another part or two. This is essentially just a tritone resolution to the key of G, which may fool the ear and make the inversion feel more appropriate.

Perhaps use the keyboard to play the C in octaves (which are inherently stable) when you arrive back at the one to offset some of the inherent instability of the inversion.

Be interested to hear what you come up with.

EDIT: Here's a resource that discusses secondary dominants in the key of C. http://www2.siba.fi/muste1/index.php?id=87&la=en