My cats are indoor only, so I've never had to deal with ticks on them, and I've never trapped any feral colony cats with ticks (and I've trapped a lot of 'em), even in ravines and wooded areas with the exception of 2 stray cats I trapped more than a year apart--in my own yard.
So, obviously ticks are an issue in my neighbourhood, although apparently not so much everywhere else in the city.
Do the ticks in your area carry Lyme disease? If so, timely tick removal is important, as the risk of transmission of Lyme disease from an infected tick increases significantly if the tick has been attached for more than 24 hours; the risk increases further if the tick stays attached for more than 48 hours. That said, I know what it's like trying to administer medical aid to a fearful, under-socialized cat, so I see where you're coming from. Most ticks do not carry Lyme disease, and chances are that Cali will be no worse for wear, but Lyme is
The other health-related issue I might be concerned about with a tick bite is an allergic reaction. Itching and redness at the site of the bite is normal to some degree, but some cats can react quite badly to tick bites. Galileo, my eldest, is very allergic to mites and ticks (common household mites, like Dust Mites and Storage Mites are arachnids, as are ticks--mites and ticks are in fact closely related, as they belong to the same taxon). We happen to know that Galileo is very allergic to both because he's had allergy testing done, but generally, cats that are allergic to mites are very likely to also be allergic to tick bites; which, if Callie does have allergies, might be a reason to persist in trying to remove the tick.
Beyond that, I would be concerned about leaving the tick because it could very well decide to detach itself and hop off while in your home. Ugh!
I have a 'Tick Key', which is a really inexpensive little tool that fits on a keychain and is specifically designed for removing ticks quickly and properly so that no blood is regurgitated back into the bite wound while removing the tick, which increases the risk of infection. It's the best method I've found for removing ticks from unhappy cats because it's quick and far less fiddly and tedious than other methods--most of which really only work if you've got a docile housecat that's content to let you pick at him with tweezers, or a whole team of cat wranglers. A vet would also be able to remove ticks in most cases, but that may not be a practical solution for Callie, considering the stress a vet visit would likely cause her.