First and foremost I am very sorry this has happened. I may be a newb around this site, but not to cars and not to BMW's. I currently have 4 and have built many motors including BMW's, the motors are purely in pro territory.
I'm sure quite a few may not comment as you know it's an advice thing..
1. I would take the 4K for parts if you can get it and afford it. I would support this by saying the rebuilding of the motor is only part of it. The putting in of the motor is a huge process. This includes dropping of the subframe, it comes out the bottom. Giant, Giant project. one thing leads to the other and the bill gets bigger. I have dropped the subframe in my E53 and it's a job! Most importantly how much time is this going to put you out? Also, your value with 110K is only salable to a certain market.
2. If you can't sell it through this mechanic sell it as a project. There are guys like me out there who might bite(i'm not looking). Disclose what happened "needs motor, otherwise perfect" A BMW shop guy might take it on and turn it(Ebay, Craigs). It's not like it has been wrecked or flooded.
3. If you decide to replace the motor consider finding a used complete motor(3K). I do believe the N55 long block could come from any N55 and the exterior parts could be exchanged(some one correct me here if I'm wrong, cams?). I've have had motors shipped from Atlanta to AZ. (Vines there has name that's been around awhile, the current reviews don't show this but worth a call, they should be able to tell you N55 exchange too.)
4. Consider
http://www.bavengine.com/other.html
5. rebuilder would have to be very reputable.
I'm amazed that anyone who builds a motor would tell you there might be debris in it and consider it ok to start. Think about how fine an oil filter is!
I was like "how the hell did that happen until I saw this:
https://autoscope.co/car-repair/bmw-...inside-engine/