to procureexcuses to a persistent client on behalf of the boss
to fabricateexcuses to a persistent client on behalf of the boss?
You would not "procure" excuses in this context - "procure" literally means "buy", although the meaning may be stretched. For example, it is possible to say "I procured a delay in proceedings" meaning "I delayed the proceedings by verbally offering something in return."
"To fabricate excuses" is a polite version of "to lie", so this would be the more appropriate phrase here.
You can "make excuses to a persistent client"; but "fabricate excuses to a persistent client" isn't idiomatic. ("Fabricate excuses" merely means "to invent excuses"; "to" on its own can't express the action of then presenting those excuses to the client.)
So you would have to put it another way, e.g.
1. He fabricated excuses to put off a persistent client on behalf of his boss.
Sad news is that the dictionaries I bought for a half of my salary are for nothing. They claim "procure" means inter alia to arrange (matters), produce (evidence), deliver or present or submit (documents).
What do you think of "produce excuses"?
My intention is to stress the fact that the excuses used before were not well-taken by clients, so it takes a lot of mind effort to invent new ones (believable this time).