I'm just unsure whether I want to make two claims at the same time or
let building insurance deal with the bigger job and I tackle the
smaller one myself.

Very valid point - I'd work out what it'll do to your premiums and for
how long. Its often cheaper not to claim.

Joist replacement is a fairly straightfoward diy job if youre a
generally handy person, read and understand the job first, and you
appreciate that the load on the floor must be supported, and in this
case jacked up slightly, by a prop (aka lengths of 2x4 about 2" longer
than floor to floor height).

And NEVER let anyone fill the bath with the joist gone! A falling bath
can kill.

If its a modern build it just might use high grade timber, such as C24,
but probably not, just C16.

Finally, if someone is genuinely incompetent they could cause a serious
accident, so
a) ensure you know what youre doing first
b) legally I dont recommend doing anything. Sue the other posters
NT