You are asking another contractor to attach to a portion of the system, the line set, that they have not installed. While it may seem a small thing, in the event of a system malfunction the new contractor ends up having to guarantee the previously installed line set. I can see where that would make them nervous. What if the line set is partially kinked? Is it completely & easily accessible.

Having been in the contracting business I will say that your proposed solution, having a portion of the job installed, purchasing the equipment on line and then asking someone to finish the install is the absolute worst type of job for a contractor to get involved with. Usually in this kind of situation, if you can find someone to do the job, as they have no real responsibility for the equipment the contractor is in no way responsible for its operation.

I would suggest that you go to the Fujitsu, Mitsubishi and Daikin sites and look at their Contractor Locator page and go from there. They are the top of the line manuf in mini-splits.

I'd also go to the guy who installed the line set and tell him straight up that you want a refund on the line-set. He will balk at the suggestion. You then tell him either he finish it or he should let his best tech (not a meathead) moonlight this job. It may be easier for him to give one of his guys the go ahead to do it and make a problem go away. This is a problem for him too.

Additionally, as I understand it, the lineset needs to be "new" with minimal oxidation of the copper... If the lineset has oxidized/weathered, then it will need to be replaced. Whether that has happened or not will depend on how they left the ends.
(and, I believe some linesets come "prefilled" with some fluid. Whether that is a protectant fluid or the actual minisplit refrigerant, I have no idea).