Similar issues in the past
We did something similar to this with one of our clients, they had something like 2,500 customers, who'd basically been warned several times that their client end would break once we'd updated security (e.g. removed deprecated ciphers etc.) at the server side (mix of FTPS, SFTP and HTTPS, this was about 10 years ago),
Customers were warned about 12 months in advance of the change, 6 months or so later, monitoring showed there were still about 25% or so of customers who had not updated.
Another 3 months and little had changed, with 20% still not updated.
Another letter was sent out, again pointing out the deadline, which was now about 2 months away. But this time we also included what we referred to as a 'live-testing', that would be done weekly, during office hours, starting about 5 weeks before the final cut off.
What this basically meant, is we'd implement the change in the live system for a few hours during the day (I think it was something like every Tuesday, 10:00-14:00), thus killing client connections for anyone who had not updated yet. We then backed out the change a few hours later. (SLA's were still fine, it would just be an inconvenience for the 20% of customers it impacted for a few hours, assuming they connected during that time window, which most did).
Needless to say, we had quite a few irate customers calling up saying our system was down, with us point out this was planned work, and that they had been informed in advance, and "By the way, this would only impact you if your system was out of date, and this will become a permanent issue in x weeks time, so you need to fix your system by then, or you'll be off permanently.".
Drastic, but it did have the desired result, with many more customers getting their updates done before the dead line.
PS: As a side note, we found out quite a few customers basically read our letters, but hadn't been understood, and not passed onto their IT people, or they had been passed on to IT, but they couldn't get their bosses to prioritise the work. Out 'live-testing' managed to focus this somewhat.