Again, it's up to the individual signing off the work to make the determination.
Buy using the guidance given in AC I pointed out written and approved by the FAA.
In the instance of the OP, if it's the same shop doing the IFR certs that will be installing the VSI I would assume they would advise their client to wait and do it all at once for the reason's cited in Mr. Levy's post #3.
Unless the test is required by the aircraft maintenance manual, the FAA does not require it either, and that was the question by the first post.
If Tom was doing the work and he believes his interpretation is correct then he can sign off the work as he sees fit.
My reasoning for following Mr.Levy's post #3 as an A&P is I believe once the system is "opened" (any part) then the appropriate test equipment should be hooked up to insure the integrity of the system has not been compromised, such as a defective instrument, broken or loose fitting, or cracked line. It's not uncommon, especially in older aircraft to crack a line or fitting when R&Ring an instrument. An I have seen instruments off the shelf have a defect. Without doing the test it would be hard to tell if the system integrity has been maintained. (As asserted in Lance's post #13 above)
While a small leak in a pitot/static system will cause errors in the non pressurized aircraft, it will make a huge difference if the aircraft is pressurized. That's why pressurized aircraft have specific maintenance instructions pertaining to opening a pitot/static line inside the pressure vessel and do require the system be tested anytime the system is opened.
YMMV.