I agree that his river range is likely either TT+, AQ, AK, or TT-QQ, AQ, AK. That range you're ok against.
However, to value bet you don't just need to be ahead of his range,
you have to be ahead of his calling range. Otherwise, it's not really a value bet (you want hands he would call with to actually be worse).
His calling range here will not contain the ace highs... and even if we assume he would bet AA/KK on the turn (which is not really certain at all here, there are reasons he would not want to), his range is still TT-QQ
That is the range you have to beat if you bet as a value bet. not the range he got here with.
Quick equity check - (you can punch it into Equilab) - there are 6 different possibilities for TT, 1 for QQ, and 1 for JJ, and we lose to all but QQ (which we split the pot with), and no chances of winning (According to equilab, you have 6.25% equity against this calling range).
So, the question remains, do we have some fold equity here? is there a possible bluff? What better hands in his range he would fold?
It all boils down to the turn check in my opinion - Is it possible he checks the turn with AA/KK for some reason here?
If not, we have almost no fold equity here, since these are the only hands that are better than yours which I think he can fold to a bet here. (well you have the 1 possibility for QQ which he might fold as well, but since you have the other 2, that has 6th of the likelihood of TT, for example).
IF, however, he can get here with KK,AA and played the turn for pot control in position for some reason (which seems like a mistake against a UTG raise in a FR table on this board to me...), for example, then betting here becomes more interesting.
However, I find 2 problems even there -
1) He's not folding AA or KK here all the time even if they somehow got here.
2) If he does, then his range on the river is TT+, AQ, AK. With this board that would leave 44 different combinations of cards.
That is Even if we assume we can get him to fold QQ-AA plus of course the bad Ace highs, those are 9 combinations for QQ-AA and 24 for AK+AQ.
That means under this assumption we win the pot by betting here 3 out of 4 times. That would indeed make it a nice spot to bet.
However, too many ifs in my opinion. IF he checks his KK/AA on that turn, IF he folds them on the river.
And again, if KK/AA didn't get to the river range, we have nearly no bluff equity or value here.
So a bet here looks iffy IMO.
I mean, it seemed like the right thing to do to me before I laid down the ranges and numbers. my instinct was to say the same thing some others have said (~40-50% size bet), went over it again, and had to edit the post before submitting after I reached my new conclusion... 
THAT'S a hand I wish we could get one of pokerstrategy's staff/coaches opinion on