Well, the custom user will allow you to create your own tables, so you can create tables and stored procedures for your gift card program. Do you have SIM/PMS licenses for your locations?
Menu changes are something that should be doable without dba access. You'll want to have your object numbers synchronized across locations and you'll need to consider touchscreen placement, SLU's, etc. It will be some work, but you'll just need to weigh the cost against using something like Product Managment or 3rd party system.
Also, I think it is pretty common to use the built in timesheets and manually enter into the payroll system. But I agree, the interface for editing clockins is a bit clunky. However, I come back to the idea that if you are repeatedly finding the need to mass edit timesheets, then perhaps this is something you can address in a less technical manner. For servers, it's usually effective to require them to show they've clocked out before they receive their tips. Late clockins can be assisted by using the scheduling tools. It's a bit of a pain to maintain schedules depending on your type of restaurant, but you can disallow clocking in 15 min after scheduled and require a manager to authorize the clockin.
That said, you may find you have write access to some of the timekeeping tables. Just be aware of the dependent tables and stored procedures.