Ruin the seed by mowing it? If its tall enough to mow--should be no problem.
If its still short--getting squashed by a heavy tire--well my opinion is that there would be almost zero damage. Accept a little extra profit. Seeding is easier than aerating. Its a good time to improve the lawn with more modern cultivars of grass seed. Darker color, drought hardy, more disease resistant. You have to convince the customer to spend a little more to go with better quality seed, of course. Naturally, in an overseed situation it is hard to determine what percent of the new seed actually took hold. And if its a shade situation there is a good chance it will become weak and thin during late summer when day length is short, temps are high and tree leaves are at their maximum.
A more important question is: What about crabgrass control? It could very well block the germination of new seed. You are forced to use Tupersan, do the crabgrass control late in the season, or treat with a post emergent. But maybe you do not apply pesticides, so you do not do the crabgrass control for this customer.