What you're creating here is a G7 sound. You're fingering is a subtle variation on a traditional dom7 fingering that alternates bass notes between the 1 and the 5 of the chord, like this.
E|------7------7--
B|------8------8--
G|-----10-----10--
D|----------------
A|--10------------
E|---------10-----
Yours just moves from the 5 to the 3 of the chord, more like this.
E|------7------7--
B|------8------8--
G|-----10-----10--
D|---------9------
A|----------------
E|--10------------
In either case, I finger the bass line exclusively with my ring finger in this scenario. In the end, there is a lot less gymnastics, over all. It will be helpful for you to visualize the whole chord and remember it as an all-purpose dom7 fingering that you can use anywhere that calls for it.
Try visualizing it something like this.
E|-----------7---7---7---7---
B|---8---8---8---8---8---8---
G|--10--10--10--10--10--10---
D|---9---9-------9-----------
A|--10------10------10-------
E|------10--------------10---
The fingering in your example is only one of many for this situation. Generally chord inversions are interchangeable as long as the melody stays correct. For example, you could completely change your fingering to something like this and it would still work sonically because the melody stays intact and the bass line is still made of chord tones.
E|---7-----3---
B|---6-----3---
G|---7-----4---
D|---5--3------
A|-------------
E|-------------
Anyhow, sorry for changing topic. The short answer is to either use your ring finger for the bass notes or to use Matt L's suggestion. The only important consideration is which one leaves you in the best place to finger the next chord in the song.
Take care!