Come for the computer mount discussion, stay for the work/life nightmare
Working wherever you are is the apparent necessity, in this case while cycling, and cycling computers aren't office equipment unless you're getting paid to ride. Pretty straightforward.
As for the actual question- Specialized are the only ones I'd trust for an answer unless they printed the mount's weight limit in the paperwork that came with the cockpit. Anecdotally, the Specialized mount seems to be the same one they use everywhere else, is listed as being compatible with the Alpinist cockpit, and includes bits to attach lights, GoPro, etc., so I wouldn't worry too much about the additional weight. The forces/load acting on the mount in your setup should be ballpark the same as having the weight out front.
https://www.lordgun.com/specialized-...r-bike-mount-1
Having already crossed the rubicon of common sense setups, you might be best served using that double-sided tape directly under the phone mount as the "primary" attachment, and then use a rubber grommet where your mount attaches to the cockpit insert so that a) you aren't preloading the insert b) mitigate load on the tape c) retain the capacity of the insert, which I'd spitball is higher than the tape.