Indeed, a bid is a bid.
You do the work, you get paid the agreed amount.
Your bid price for the work can come from your flat rate sheets or from doing a detailed material takeoff and estimate the costs and adding a markup (if you have the time for all that). If your labor costs are a little higher than someone elses, for whatever reason, your flat rate sheets will contain slightly larger numbers.
Your flat rate sheets should be based on your averages to do specific work tasks, weighted slightly in your favor (contingency factor?), and should be enough to cover all direct costs (including you), indirect costs, overhead markup, plus something for company profit.