I formed a single-member LLC (taxed as an S-corp) in Texas for online technical consulting. I lived in Florida temporarily for 6 months to work at a client's office and then moved back to Texas, where I have a permanent home.
The payroll should be reported in Texas, as that is where the company is located and where you work full-time. That may be better for payroll tax purposes as you would probably pay more for state unemployment taxes when registering a new business in Florida, as there wouldn't be any favorable history of unemployment claims to reduce your unemployment tax rate. Neither state has a personal state income tax so that wouldn't be a factor.
You may be able to deduct travel expenses since you are away from your tax home overnight but for less than a year. You would have to justify why it was necessary to rent the apartment, why you couldn't do the job from Texas in this case. Also, the lodging could not be too lavish or be used for primarily personal purposes. For instance, you can't take an extended vacation and just do a little bit of work and claim it was a business trip.
Technically, the lease should be in the name of the LLC. You may be able to get around that by issuing expense reimbursement reports showing the lease expense if the apartment was in your personal name.
I don't think it would make any difference from an IRS scrutiny standpoint to choose a mileage deduction versus deducting actual business vehicle expenses.