Am I incorrect in thinking they also can’t afford to let rates rise too much as they can’t afford the interest payments?
But yes, raising too much too fast will cause stress and defaults, and nobody wants that unless it's absolutely necessary.
Fed tightening would force fiscal side to actually restrain spending which would help alleviate inflation. That's the whole point of tightening to begin with (tightening both public and private credit).
The reason the Fed is supposed to be independent is to avoid a Venezuela type situation where deficit spending is monetized through money creation. Well the Fed has been effectively doing that through QE for two years.