AQ in SB (Bubble Play Tourney)

[D-Nut]

Now this one is more of a tournament theory question than an actual hand play question.....

Here is the hand first:

Blinds 250-500-50

(Now let me preface by saying that the tourney is the 50/50 on FTP, and my tables have been stacked with very good players and even some big names)

I am in the SB with roughly 12500 (average chip stack)
Villian has roughly 15K

The hand folds to the cutoff who raises to 1250. Folds to me in the SB, and I look at

I 3-bet to 3250....back to the cutoff who pauses and raises me all-in. I call.....villian shows 4-4. I flop air....turn a queen of spades...he rivers his one outer, and I'm on the rail.

Now....here is where my question lies. When I 3-bet, I subconsciously left enough fold equity for my villian to shove all-in and think he could make me fold.....what ultimately happened is that I created a race situation, that may not have been created if I had simply shoved all-in.

Now....my issue with bubble and mid stage tourney play is that I used to show down NO hands....I'd fold to pressure and widdle away and cash just beyond the bubble......NOW, fast forward to now....I'm showing down TOO MANY hands....and as we all know, you showdown too many, math dictates that you're gonna lose your fair share.....so I am thinking it is important to pick up non contested pots, and thats not what I'm doing (This hand as an example)

So.....did I make a mistake by 3 betting short here?