What you want is "I've been busy since the moment I arrived", "I've
been busy since I arrived", or "I've been busy since I got here."
We do not have to use the same tense in the main clause and the
subordinate clause. Certainly not in the general case. In
the specific case of "I have been X since Y", the event described at Y
is almost always going to be in the simple past. I recommend
memorizing this pattern because it comes up in hundreds of ways all the
time.
I've been tired since I began taking English lessons.
I've been happy since I fell in love.
I've been sick since I took those pills.
We've been constantly cold since the furnace broke.
He's been very tan since he started lying in the sun every day.
They've been very friendly toward us since they learned that we were rich.
This is one of those structures for which it makes no sense to have
both clauses in the same tense. The present perfect shows a
situation continuing into the present; the "since" clause has to show
where (
in the past) the beginning of the situation occurred.
CJ