I am not a handy guy and have replaced the alternator on an older car. I cannot imagine the Elise is too diferent. Really was just several bolts and then trying to get the stupid belt aligned properly. $1700 seems extremely over priced! That said, I have no clue how hard it is to work inside our tiny engine bay...
drop the belly pan and take a look up to the USPS side. there's not a whole bunch of what anyone would call room up there. it's very tight.
here's a short thread which discussed the serp belt and tensioner:
also, to the OP, i would suggest that you at least consider replacing the belt while this work is being done. perhaps it's included in the estimate?