If a used car dealer will not allow an independent inspection in any reasonable form, I would keep shopping. Before reaching that point, though, establish what the dealer is refusing. “You cannot take our car off the lot for the afternoon” and “You cannot have your mechanic inspect it” are different answers.
The first may be a problem you can solve together. The second asks you to accept a purchase without information you have decided you need. A good price does not fill that gap.
Make the request concrete. Name the shop, explain the proposed appointment and expected duration, and tell the dealer who will pay. “My mechanic can inspect it at 10 tomorrow and needs about an hour” is something a manager can approve or work around. An open-ended request to disappear with the vehicle is harder to arrange.
The FTC identifies two alternatives when off-lot access is restricted: a mobile inspector can visit the dealership, or the dealer may take the vehicle to the buyer's chosen facility. Either could resolve the access problem without giving up an independent opinion.
Speak with the inspector before settling on the location. Explain the car and the questions you need answered. If the proposed visit cannot include part of the inspection that matters to you, find out what that leaves unresolved. A convenient appointment has little value if everyone discovers its limitations after you have paid for it.
For example, you might be satisfied with an initial on-site assessment before deciding whether to pursue a more involved inspection. Or your technician might explain why the car needs to visit the shop before a meaningful opinion is possible. Let the work you need determine the arrangement rather than booking whichever service is easiest to fit into the afternoon.
An inspection appointment also needs agreement about the sale while you wait. Is the car being held? Is a deposit required? Under what conditions would that deposit be returned? Put any agreed conditions in writing before paying. Do not assume booking a mechanic automatically reserves the vehicle or creates a right to cancel a separate agreement.
When the dealer has already inspected it
If the salesperson offers the dealer's reconditioning report, read it. It may answer several questions and help your mechanic focus the remaining work. There is no benefit in treating useful documentation as worthless simply because the seller supplied it.
It still serves a different purpose from the independent assessment you requested. The FTC recommends independent inspection even for a certified used car. You can appreciate the work already done and still want a technician you chose to evaluate the purchase for you.
What would concern me is a refusal of every practical arrangement, especially if it comes with pressure to sign immediately. You do not need to prove the dealer is hiding a defect to decide the available information is insufficient. Buying another car is a reasonable response to an unanswered question.
Keep the conversation calm and specific: “I am interested at these terms, subject to an inspection we can arrange. If we cannot arrange one, I am not ready to buy.” That gives a willing dealer room to solve the logistics and gives you a clear point at which to leave.