score:7
Not a problem. I've taken all sorts of weirdly shaped items in the past (including a full stereo system in shopping bags!), and as long as you meet their criteria, ie:
then you'll be fine taking it aboard.
Saying that, measure the backpack first - I'm taking a 30L pack on a flight again today and it's actually small enough to fit into most airlines' carry-on limits.
EDIT: Now that their site is working for me, I can see there's actually a FAQ about this on that link above:
I carry a big purse. Will I be charged a fee to bring that on board or can it count as my personal item?
Which sorta covers your question - basically will something large be too large? And they respond..
All carry-on and personal items must fit within Allegiantβs luggage dimensions and may be verified prior to boarding. Please measure your bags carefully, including exterior wheels, pockets, and decoration.
Personal items which do not fit completely underneath a seat may be considered a carry-on bag and passengers will be charged accordingly. Excess and oversize bags may be gate checked.
The personal item must not exceed exterior dimensions of 7 in. x 15 in. x 16 in. (17.8 cm x 38.1 cm x 40.6 cm) and must be placed under the seat. The carry-on bagβs exterior dimensions must not exceed 9 in. x 14 in. x 22 in. (22.9 cm x 35.6 cm x 55.9 cm).
So as expected, as long as your 'personal item', whatever it is, meets their requirements, you should be fine.