It'd be very strange for a client to trigger a 500 error on the server, I really would recommend finding out exactly what the error was as it may aid with getting to the bottom of the problem; currently in the event it turns out to be VM there isn't much for the network team to work with, as your traceroutes are fine.
If the problems are caused by the same issue it's best to focus on your own site as you have more control over viewing the server side of the connection and obtaining more in-depth information that way.
I can't see the images until they're approved by a moderator, however I've taken the opportunity to do a little searching about the ebay issue. I can see the mentions of prior issues with ebay from 2015 and 2017 but I can't see a mention of an issue on the ebay discussion forums or elsewhere of any issues as of late. The past VM issue seemed to be down to routing between ebay and VM and as such affected the entire site and not just the account panel since the different parts of the site use the same route.
EDIT: Right as I posted this the images got approved, that is indeed strange that it's hanging like that. I've checked back over the traceroutes you've posted and whilst the ebay ones do look fine, both the one whilst you were backing up and the one after to your wordpress site are showing packets being lost after the connection enters the namecheap network, and in the traceroute from after you've finished the backup it even seems to enter a routing loop and hop 11 and 14 are both the same.
You may need to contact namecheap and see if they're aware of anything or can perform a traceroute to you from their end (if the option isn't available for you in cPanel.)
Here's is a traceroute to the namecheap wordpress site from an OpenReach/Plusnet connection. Which as I said the site works fine on all day and every day.
1 1 ms 1 ms 1 ms dsldevice.lan [192.168.1.254]
2 * * * Request timed out.
3 * 7 ms 7 ms 141.hiper04.sheff.dial.plus.net.uk [220.127.116.11]
4 7 ms 7 ms 7 ms 140.hiper04.sheff.dial.plus.net.uk [18.104.22.168]
5 8 ms 7 ms 7 ms 22.214.171.124
6 13 ms 8 ms 7 ms 126.96.36.199
7 8 ms 7 ms 7 ms 188.8.131.52
8 7 ms 7 ms 19 ms 184.108.40.206
9 * * * Request timed out.
10 8 ms 8 ms 7 ms ae-1.r05.londen12.uk.bb.gin.ntt.net [220.127.116.11]
11 8 ms 7 ms 8 ms 18.104.22.168
12 * * * Request timed out.
13 11 ms 11 ms 12 ms premium46-1.web-hosting.com [22.214.171.124]