To directly address OP's question and not offer other advice about airlines and packing tips:
I have shipped to and from BCV.
I have not done it since COVID, so forgive me if the procedure has changed. When I did it, anything shipped UPS or FedEx was delivered to the Yacht & Beach Club conference center's business office. Retrieving the box from the conference center was a bit of a pain in the neck. You can call the conference center business office and ask them to have someone bring it to your room. That might work, if they have an employee around who is willing to do that. You can walk over to the conference center yourself, if you will be able to carry the box yourself. The only problem with doing that is if they only have one employee around and that employee decided to deliver a package to another guest, so you get there and nobody is working in the business office. Also, the conference center business office is only open certain hours. If you want your package before or after that, it might be closed. If you are in that situation, you can try going to the Beach Club front desk and asking them to have someone go over and get the package for you. I did that at BoardWalk but I haven't tried it at Beach Club.
One time I arrived and my room wasn't ready. I walked over to the convention center, retrieved my box, and gave it to Bell Services to hold until my room was ready. That way, I knew where it was and that I could have Bell Services bring it up to my room with my luggage when I was in the room.
It's not a long walk from BCV to the conference center, but I seem to recall reaching a point where there are no signs and I wasn't exactly certain where I was supposed to go. The walk back from the conference center to BCV was confusing. One time I accidentally ended up walking over to the Yacht Club. I think I went out a different door from the one I used to enter the building.
The business office is supposed to charge you a fee for receiving your package. Use the link below to see the fees.
The business offices are not run by Disney. They are run by an outside company.
I shipped FedEx, not UPS. I only did that because the FedEx near my house is easier to get to than the UPS. The first time I did it, the FedEx employee happened to be someone who had family in the Orlando area to whom she had shipped stuff in the past. She said my box would get there fast even if I didn't choose overnight delivery. I think I paid for cheaper 3-day delivery and it arrived the next day.
I also have shipped stuff home from the Yacht & Beach Club convention center. I had packed a clear trash bag. I put the stuff I wanted to ship in that, and took it over to the business office. I did not have a box, but they boxed it up and shipped it home for me.
Oh, yeah, another tip I read on here years ago: have the kids decorate the box with crayons. Or put your name on each side of the box. One time I checked the FedEx tracking and saw that my box had been delivered. But the person in the business office looked and said they didn't have it. I made them look again. My name only appeared on the top of the box. They had set another box on top of my box. So when they glanced in the back, they didn't see a box addressed to me.
Here is the link to the info about the conference center business offices:
https://disneyworld.disney.go.com/guest-services/business-centers/