Here's how I resolved it, and here's what the problem turned out to be.
TLDR: If you have a business PayPal account, fill in each and every possible section of the Settings > My Profile > My Business Info panel that the panel will let you complete (in my case, 'Trading Name' is greyed out, for instance - PayPal neither autofills that information from my online info for the company, nor lets me complete it manually). Otherwise there is an endless string of ceilings that you will hit as the cumulative amount of money passing through your account rises. AFAIK the details of these ceilings are undocumented or otherwise unavailable to the public. Just fill them all in on day 1.
You'll have to really look for them, too: click and explore each 'Edit' link in that section, and don't be fooled by what the labels are on those links, because they are completely unintuitive. Nor will the interface (web or mobile app) be able to tell you what information is missing, or what section you should go to to fix the problem - only a live person on a help call can do that.
In desperation at the opaque messages from the web interface and the PayPal app, I hit the 'Call' button, not even knowing if I had enough international phone credits to call the US from Eastern Europe. Turns out, I did.
After less than a minute I got through to a lady who was probably as helpful as anyone could have been under the circumstances.
The PayPal account in question is a Business account, and, as far as I can tell, this new payment of $1000+ pushed the account into a new level of required authentication.
The lady was able to see which missing information in my account was causing the block on transactions, even though this is not indicated to the end user in any way while he or she is blocked.
The lady directed me to Settings > My Profile > My Business Info > Business Information. All that field shows in the panel is the name of your company, which might lead you to believe there is only one editable field in there. In fact, there are a great many fields in 'Business Information', and the fact that they were not filled in caused my account blockage once we hit a certain ceiling of money throughput.
The lady told me that to unblock the account I had to add a business phone number. We had already added one, a mobile for my partner, who has the majority share in our company. I tried to edit the category of that number into a 'business' number, but there is no such category available. There is only 'Work', 'Mobile' and 'Call center'.
In the end the lady advised me to re-add the same number in the category 'Call Center'. After I did that, she confirmed from the back end that this had lessened, but not removed the blockage.
Next I had to add a 'Beneficial Owner' in that section, so I added my majority-holding partner, who has 66.67% of the company. The lady said that I did not have to add myself straight away to unblock the account, but that eventually I would need to do this. I decided to wait until I had withdrawn the money that I wanted to withdraw, and afterwards I added myself, anticipating later blocks when our throughput hits yet another undocumented ceiling.
I suggested that I fill in the VAT no. section, but the lady said not to bother. I am pretty convinced, based on experience, that this account is going to freeze up again at a later point for lack of a VAT number, but nonetheless I heeded her advice.
I also had to insert a 'Sales Venue' (I just put 'LinkedIn', which is where my partner and I mostly get our contacts and work), a pretty meaningless 'Average transaction price', 'Average monthly volume' and 'Percentage of annual revenue from online sales' section (could not complete the more relevant information without putting in some or other random figure for these, even though they do not really apply).
After that, the lady said that the account was unblocked, and I was able to transfer my money.
I have not had a bad experience calling for help with PayPal, though I don't know if they prioritize foreign caller IDs. The lady was helpful.
But what burns me is that the account locks up without providing any feedback or route to a remedy, or email notification of any kind, anywhere in any of the supplied interfaces. Combine that with the fact that PayPal apparently never replies to emails (I sent two during this problem), and you would really have to Google the problem and find out in a forum like this what might possibly be wrong with your account.
