Income is income is income, be it from markup over cost on tires or having something for a tech to do in spare moments. Everything is overhead (building space, utilities, labor, taxes, etc) until overhead is paid for, THEN anything left is profit.
With a machine, mount and balance on two wheels should be what- half an hour, tops? Using the previously mentioned $30/wheel rate (off the bike), that's $120/hour. What's the tech make? Double that to cover burdened costs (insurance, taxes, your mentioned "overhead") and it's still 50% "profit".
Or you can let that tech stand there scratching his nuts for that half hour, generating no income for the shop.
Worse is "I don't want your business". That's messed up. Tell them "our insurance coverage won't let us mount tires brought in from another supplier", which is BS but at least it saves face. Tell him "OK, this time, and next time check, we'll price match what you paid online" or "if you buy them here, we'll do the work for less" is better. Telling a customer
(that is, a person with a desire to give you money) to go away
is a shortcut to bankruptcy.