Me, my problem has always been eagerness to please, so I'd agree to "just do it" and skip the first part.
Don't, under any circumstances, do that.
If the client asks for a fixed price, then don't propose an implementation cost. Instead, propose a discovery cost. I can easily say how long I'll need to figure out what needs to be done and deliver a documented solution with an estimate of the work needed.
If they refuse this approach, then the only other option is "time and materials", i.e. I will send you a log each week of the hours I've worked and the product I've delivered, and you will pay me each week.
If they refuse discovery as well as time and materials, and insist on fixed price, then I walk away.
Don't, under any circumstances, do that.
If the client asks for a fixed price, then don't propose an implementation cost. Instead, propose a discovery cost. I can easily say how long I'll need to figure out what needs to be done and deliver a documented solution with an estimate of the work needed.
If they refuse this approach, then the only other option is "time and materials", i.e. I will send you a log each week of the hours I've worked and the product I've delivered, and you will pay me each week.
If they refuse discovery as well as time and materials, and insist on fixed price, then I walk away.