I agree with this completely. It's all about the engineer.
Not only will you get the advantage their skills when recording and mixing, but you'll also get the benefit of their advice every step of the way. If they are truly on top of their game that will have a positive effect on your production in countless ways both big and small.
That said, I think you'll find $10,000 to be a very tight budget. Realistic budgeting is important because you don't want to get half way through the project and then run out of money. That happens all the time on independent films for example, and it can delay the project by however long it takes to get the rest of the money together.
The best thing to do would be to figure out to the best extent possible where *every dollar* is going, before you start. I'm talking everything: engineer, studio, cartage costs, storage (i.e. hard drive or tape), what the tracking will cost, what the mixing will cost, what the mastering will cost, airfare, taxi, food, hotel, etc., etc., etc. I mean, do a full budget and then you'll have a better idea of how to proceed.
$650 a day for an engineer on a $10,000 budget doesn't sound realistic to me. If the band knows the music cold, and has been performing it regularly, then tracking wouldn't take so long because you could just go in and lay it down. So it's possible. But if tracking is going to be a "creative" process and the music hasn't been gigged much, then tracking alone could easily exceed the $10,000 budget.
You really need to get down into the nitty gritty of figuring out that budget.