You absolutely need to look at the cockpit and frame together, the frame size you need could well change depending what cockpit you want to put on it. And with the more integrated “superbikes” you pretty much have to use the cockpit it comes with.
The way I would approach it is to get your current stack and reach numbers, and also your fitter’s best view on how they might change over time as you adapt to the position, improve flexibility, etc. E.g. if he thinks it would be beneficial to get lower over time, then you need to make sure you’re on a frame/cockpit that will allow for that. Shortlist the bikes and cockpits you’re interested in, then go to their respective websites and most of the good ones will have some manner of fitting table where you can plug in your stack and reach numbers and see which bike sizes will fit.
Here’s a good example from TriRig (I have no affiliation with TriRig, just like their products and how simple they’ve made the fitting process for their cockpit) - https://www.tririg.com/store.php?c=alphaone&page=fit You can choose the frame you’re looking at, look up your stack number, then see what your options are for achieving that stack i.e. how many stem spacers and how much pad stack.
Similar link for Zipp cockpit - https://www.zipp.com/support/vukafit
Or Cervelo P5 - https://www.cervelo.com/en/p5
No stem spacers is a good goal, bike will look slicker and likely be more aero. You will want some spacers to raise the pads above the base bar (or pads that are raised by telescoping instead of spacers, such as the TriRig or Cervelo ones). Having some pad elevation above the base bar means that when you are on the base bars you have a lower and more aero position (may or may not be useful depending what kind of riding you’re doing), more importantly it gives you the option for dropping your pads to get a lower front end in future. If you have a slammed stem, and pads that are sitting right on the base bars, then you have no options for lowering the front end further, unless you can fit a negative angled stem which starts to look pretty quirky.
Other factor to discuss with your fitter is whether he thinks tilting your arms is a good idea. Seems to be all the rage these days, a lot of people find it more comfortable having a slight upward tilt which helps to lock your elbows into place, there is some evidence it may also be more aero. If you want the option to tilt, either now or as something to experiment with in future, then that may also influence your choice of cockpit (or bike with an integrated cockpit). Some cockpits don’t natively support any tilting and you’d need to achieve tilt by adding parts from another supplier such as angled spacers/wedges. Some cockpits support tilting but with fixed increments (5 degrees, 10 degrees, etc) using their own parts. Some allow you to adjust tilt without using spacers so you can have whatever angle you want.
Sorry if this all sounds a bit complicated, the challenge with TT bikes is that you’re locked into one hand/arm position for pretty much the whole race, so it’s critical that your pads and extensions can be put in exactly the place you need them relative to your saddle and pedals. Whereas on a road bike you can shift between hoods, bar tops, various different drop positions to get the right balance of power, aero and comfort depending on whether you’re sitting in the pack, climbing, descending, etc. The good news is that manufacturers have mostly recognised the need for a lot of adjustability, so most integrated bikes and standalone cockpits are pretty good on that front now, certainly compared to a few years ago.
The other good news is that you’ve done exactly the right thing going to a fitter first to figure out your position. Rather than buying a bike because it looked cool or was on sale, and then trying to work out how to fit yourself on to it after the event, which is all too common!