I have a 1985 K100rs

Long story short had a problem that I took into my locally owned BMW motorcycle repair shop and they have diagnosed that it needs a new motor and to get one off eBay. The problem is my bike has been customed and while nothing about the engine/transmission is custom, they don't want to do the repair because it would take time away from other businesses they would have to turn down. They suggested I try to find someone else did offer to try and take on the project in another month or so, but I live in Seattle, WA, and will essentially lose the chance to ride until next Spring due to poor weather if I don't find someone else.

Can anyone recommend someone skilled enough in the Greater Seattle Area that could replace a motor?

Piece of cake, except I'm not really near

Sorry to hear about your problem - and sorry to hear of a BMW shop turning their nose away from an older BM.
Dunno if you've ever encountered a Haynes workshop manual, but they have a system of awarding 1 to 5 'spanners' to a repair job according to its difficulty (5 is the hardest). I'd give an engine swap about one spanner. It's just five big bolts, the rear shock, a load of no-need-to-worry wiring connectors, and then you just wheel the frame and front wheel away somewhere leaving the transmission on a milk crate (or equivalent prism of your choice). 
Post some pics, and everyone on here will enjoy talking you through it. Even Arlina can do it  .

I'm assuming you've got basic tools and somewhere to do it. Apologies from me if you haven't. Good luck.

What was the diagnosis that led them to suggest a replacement motor?

MIke

I have never done an engine change on a K....but been in there enough to agree on the Haynes one spanner rating. Sadly if you go to the main dealer its gonna cost more than the value of the bike but in fairness to him he gave you solid advice and saved you a lot of money. If you have a mate and a little knowledge about Ks its go to be in your skill set. Might take a bit longer than the book but its not so difficult. 5 bolts and lots of electrical disconnections that are generally different plug and socket connections that are not so hard to reconnect. Is your local shop a main dealer or independent specialising in them?

Outside of wrestling with a heavy lump of metal, an engine swap isn't all that difficult.  I have found that it's easier if the bike has had the wheels and fairings removed.  It's amazing how few tools are needed.  I would guess that less than $100 would get everything you need at Harbor Freight if you were starting with an empty tool box.

If it's a new to you bike, an engine swap is a great opportunity to go over the bike in detail correcting years of wear and bringing it into reliable operating condition.  Having the engine out of the frame allows you to clean and service every part of the bike getting it to a great starting point for your ownership. 

Things like touching up the paint on the frame, replacing perished rubber bits and cleaning electrical connectors and switches.  Jobs like the steering bearings, rear main seal and clutch spline lube are easier on a bike that has been stripped down.

There are threads here and at other forums on the interweb documenting the refurbishing of of other bricks.  These threads can provide a lot of practical information that will reinforce your confidence in taking on the job.  Along with a Clymer and the download factory manual, the members here can provide all the help you need to do an excellent job.

These bikes are a dream to work on. Engine swap sounds drastic but I can’t imagine any bike I’d rather do it on. Good luck.