I'm in two minds about the comma and the "but".
Yes, "but" is a "joining word" ... a conjunction.
In this context, it's a "co-ordinating conjunction" (joining two parts of the sentence of equal importance), and cc's are often prededed by a comma.
eg. "We could discuss grammar now, BUT perhaps it can wait until morning."
(as distinct from sub-ordinating conjunctions which link a main clause and a sub-ordinate clause ...
eg. "I am tired BECAUSE I spend so much time on AB"
... where, I agree, the comma should be left out.