I'm not an expert, but I have been a 1099 contractor for many years and researched the topic fairly extensively. I would say you would have a hard time proving to the IRS that you are a private contractor--at least in this particular case. You are working his schedule, using his equipment (vehicle), etc. That sounds like an employee to me and it sounds like the 1099 contract is just an (illegal) excuse not to pay employment related taxes. Here is a quote from the IRS website (via a secondary website):
The general rule is that an individual is an independent contractor if the payer has the right to control or direct only the result of the work and not what will be done and how it will be done.
There are likely other rules that would make your mileage deduction illegitimate.
If you were setting your own schedule, driving your own vehicle to the site, using you own tools, etc. Then you would likely be considered a sub-contractor and could, indeed, claim mileage, even if you had to show up at his shop to get materials, job information, etc.
I hope this helps
