There is no consistent theoretical basis for electing an at-large senator by one method (STV), with one voter base, and then switching to a different method (FPTP) and different voter base to replace that representative in the event of a vacancy. It does not do the job of making sure those voters are represented, as the goal for special elections ought to be. That is done far better when party chairs can appoint an ideologically similar replacement that preserves the will expressed by those voters.
The Senate seat would be filled with the RCV voting system, not FPTP. Also, a counter argument I'd like to make is that the people elected them, and even if they were replaced by someone who was "ideologically similar", the replacement wasn't chosen by the voters who voted for the person who resigned.
I'd be fine with a system where if someone who put the Senator who resigned in their top 5 preferences or something like that, they can vote to replace the Senator in the special election.
Also, the regions of Lincoln and Fremont have a system like this and nobody in recent memory has complained about that.