That's what the TMSI is supposed to be - (iirc) older cell protocols would send the IMSI over the air all the time, but since at least LTE, possibly earlier, the IMSI is only involved during initial connection to (some region of) the network. During the initial connection, you're assigned a TMSI, which then changes periodically, and you use this for identification on subsequent transmissions.

This is why this paper also talks about determination of IMSI from TMSI.

https://en.wikipedia.org/wiki/Mobility_management#TMSI

But your idea is plausible as an added layer of security - the SIM card could generate a temporary IMSI somehow. I think you'd actually want the next IMSI (or next N IMSIs) to be assigned by the network and communicated to the phone/SIM, rather than deterministically generated. If you deterministically generated the numbers, you'd have to pre-assign each subscriber a pool of numbers that are guaranteed to never overlap with any other subscriber, and this scheme would have to:

  1. assign a large enough pool of IMSIs to each subscriber to handle all future communications with the network
  2. make it so it isn't trivial to determine whether two fake IMSIs are the same subscriber (so you can't do something like a fixed prefix + HOTP or something).