Check with your own Rabbi, but as I understand it, the role of the shadchan is broken into three parts.
(1) Make the suggestion.
(2) Handle the dating situations as the come up.
(3) Smoothing out any last minute bumps in the road that gets the couple to engagement.
If a shadchan pushes you to meet or speak without checking references, when you are clear that you prefer to do so, or if a shadchan tells you to lighten up when you are trying to be clear about the boundaries that make you comfortable, or in any other way doesn’t seem to be allowing you to make your decisions about how you like to date, put someone else in the middle. I encourage everyone to network with many different shadchanim and shidduch groups. When you work with a shadchan that you feel “gets you” and that you are comfortable working with, ask them, “in the future, if someone else makes a suggestion but isn’t a shadchan or I am not comfortable using them as the go-between, would you be willing to step in for me?”. That is what I call the “Shadchan Shuffle.”
You have to be careful not to upset the person who made the original suggestion and not to seem to the other party in the shidduch to be a micro-manager.
I recommend just saying that there is a person who you work with often and because that person knows you better than the person making the suggestion, you prefer if that person handles the shidduch from here.
Another option is to take the suggested name and say that you will get back in touch if there is anything to discuss. You can then have your preferred shadchan get involved and be in touch with the original person if the shidduch ends or if you get engaged.