The most established way to handle a price objection is to ask a question. It goes like this:
“If we were the same price as the competitor would you prefer our proposal?”
What is the client to say? If they say “Well, no actually,” seemingly the worst outcome, at least you can now ask, “Why?” and deal with the other objections real or imagined. Suppose they say, “Well, yes I think we would.” At that point the same question, “Why?”, will elicit what the client believes to be special about your firm. Often clients are actually trying to persuade themselves that the bit extra is worth it.
In my experience very frequently the question, “Would you buy from us if we were the same price as the cheapest?”, leads to a lot of thought from the prospect. Keep your mouth shut and wait for the response.