Road bike to triathlon bike - bike handling or fit issues?

Why would you want to do this anyway? After all, according to the comments on this story http://road.cc/content/news/243808-police-probing-triathlon-cyclists-undertake-horse-rider a triathlon competitor isn't even a Real Cyclist (TM)   

Nothing of use to add here, apart from stating the obvious that if you are putting your weight nearer the front, more than the rear it will cause more twitchy steering.

On a road bike I think the ratio is about 60/40 rear front weight distribution, on a Tri bike I'd suspect that changed to have a bit more on the front.. might be worth weighing your position on the bike using some bathroom scales, it's possible you're too far over the front, or maybe that'll be your new normal on a dedicated TT bike?

My cousin let me borrow his Quintana Roo TT bike for my first Tri I did about 6 weeks ago.  I did feel a little less comfy/stable on it than my road bike with clip-ons, but part of that was due to the bike being just a tad small for me.  My legs are longer than my cousin, so I had to move the seat up a few inches to get my normal leg extension, which put me into a fairly aggressive position (cousin already had max spacers under stem).  It also meant that I felt much more compact than on my road bike - granted, my bike is slightly big for me

Because of how compact I was, I felt like I was constantly in danger of falling forward, which made me more tense and probably contributed to how unstable I felt.  I'm not sure about your stem/handlebar setup, but my cousin's bike has a traditional stem holding the TT bars.  So for me, angling the handlebars up slightly seemed to kind of push me back towards the saddle, which made me a lot more comfortable and even made the ride feel more stable - possibly getting my center of gravity back a bit?  Still wasn't as smooth as my bike but I felt much more relaxed, which can definitely help in feeling more balanced.

It took me a while to get used to my TT bike, having never used any type of aero bars before.

I was fine after around 100 miles. The handling is still absolutely atrocious though due to the steep frame angles.

I don't think it's simply lack of practice, as you've already ridden with clip-on aerobars.

A tri bike shouldn't be any less stable than a road bike with clip-ons, even if the position is a little more aggressive (i.e. lower at the front). If anything, tri bikes usually have slightly more stable geometry to provide less twitchy steering, though I don't know how the P3 compares to the S5.

I'd go back to the shop where you were fitted and ask for advice. Have you compared the position of the saddle and bars between your two bikes? Are the saddle height and fore-aft position correct?

Also, I wouldn't make fitting/position decisions based on photos of pro riders. I wouldn't even compare myself with the people I race with regularly - we are all different shapes and sizes, on different sized bikes and with individual preferences and needs.

cervelo has a steep head angle making it have faster steering. TT bikes are less stable than road bike due to width of steering. They take practice to develop the handling skills, technique. They will always be prone to crashing.

my limited experience is I ride further away from the kerb to allow for swerving. Also on unknown courses with lots of bends, get on the brakes early, sod the speed, save yourself.

Looking at all the Pro's most have upper arms either perpendicular or slightly forward. But I guess setup is individual

I suspect you either get on with tri bars or not. I took the plunge, got a dedicated tri bike to do 10 mile TTs but never felt quite secure even when the road was straight and in perfect condition. Crashed on my 6th outing, broken clavicle over 8 weeks ago, still not allowed to ride outdoors, so the bike will be sold. Possibly need to be younger and braver when you first try it? Im no expert but thought I'd read in CW about getting a decent aero position without tri bars, but maybe harder to sustain for longer durations.

Nothing much useful to add, but as a road cyclist I've always wondered how anyone feels any semblance of control on tri-bars.  Then again I don't even like descending on the drops so maybe I'm just a wimp