You have your concerns... and I want to try to reassure you that most of what you are worried about in your post are mostly minor.
The modern printable vehicle magnets are soft and very flexible and come to the manufacturer on huge rolls instead of rigid sheets. I am familiar with companies using magnets for partial and full vehicle wraps (temporary or seasonal) and they actually protect the vehicle better than a custom paint job.
It is highly unlikely that they would directly cause any damage to your vehicle. If you are still concerned, ask for a design that includes rounded corners instead of a straight rectangle cut and there will not be any sharp edges on the magnet sign to begin with. Think of the flexible magnets that go on fridges, and imaging how difficult it would be to cause any damage to your paint with one of those.
As far as damage from uneven sun wear, you can easily remove it once you leave the job site. They are easy to attach and detach with no more effort than peeling a banana. If you travel to this job site 3 times a week for a few hours at a time... 10 hours of sun exposure during 77 hours (winter) or 90+ hours (summer) per week will make very little difference.
Also does the contract require that you have a magnet sign on both front doors of the car, or is it something that you only need to have on the driver side? (Requiring on both doors makes it more of an effort for you, and its possible you get distracted and forget to add/remove the opposite driver side sign).
Is it something that you would be able to first park and then affix the sign to the door, or does the sign need to be in place before driving onto the property?
For me the bigger concern would be if the sign is stolen, vandalized or falls off.
As an employee are you responsible for the damage/loss?
When you requisition the signs from the boss, recommend that you get 4 copies of the sign even if you are only required to put on one on the drivers side door. Explain that because they are easily removable, they could be lost or stolen and that it is better to have extras in the trunk then to show up on the job site and be refused access because the sign falls off during transit with no backup available. Explain that you think that 4 should be sufficient to last the entire year. (What you want to do is set the expectation that these are "disposable" and not valuable company assets that you need to be held 100% accountable).
Also, it is not really all that much more to order 4 custom signs than it is to order just the 1.
In fact the cost to get custom signs is less about the material and square footage cost to print and more about the setup and custom art. In printing, the cost per item always goes down the more you order... but if you order fewer than what you actually need then you end up spending considerably more in the long run.