Eliminate the fixed size of the court entirely. We don’t need to define 9 or 15 people.

Every presidential term, the president appoints two new candidates, 9-15 months after the presidential election, and 9-15 months after the midterms. We do not fill any vacated seats.

That resolves the problems with multiple seats unexpectedly swinging on a small court, and limits the effects a single president can have on the court. Yes, the numerical swings can be as large, but the percentage swings will not be; the court will likely fluctuate between 15-20 justices.

Now to fix the Senate playing games…

First, we establish a line of succession with the circuit courts. The chief judges of the circuits, in line of seniority, then every other active judge. Every case before SCOTUS requires at least 6 justices to hear the case. If the court falls below 6, the next judge in line is automatically elevated to the court. If the court is larger than 6, but due to recusals or abstentions, fewer than 6 are able to hear the case, the next judges in line are automatically, but temporarily elevated to hear that case. Only when we have exhausted all judges from the district courts does the president get additional, temporary appointments.

Any appointment to the circuit court requires senate confirmation. After we enact this, any judge confirmed to a district court could (eventually) find themselves on the court. Their confirmation thus includes the (remote) possibility that they will be elevated to the court. So any circuit court appointment after this goes into effect also serves as a SCOTUS confirmation.

When it comes time for the president to appoint a candidate to SCOTUS, anyone who has previously been confirmed to the line of succession can be immediately elevated to the court, without needing additional confirmation.

Where the president and Senate are sympatico, the president can choose anyone they want. When they are at odds, the president still has a list of pre-approved candidates the Senate can’t block.