Have a look at Clook who are based in Preston (so you may get a Lancashire accent if that passes for English ...lol), one of their larger Reseller packages should fit your needs, though you'll need to contact them to get specifics as they don't advertise them on their site, in fact their Reseller-4 (£45/mo) appears on the cart/order page - https://www.clook.info/clients/cart.php?gid=3 - which may fit?
Clook have one of the best reps in the industry for customer support, they don't have a live chat but all tickets are answered in minutes, my fastest ever response was just 20 seconds. I've been with them for over 10 years and rarely need to use support, the only regulars are SSL cert updates which they turn around usually in 5-10 minutes or so. They monitor all their servers 24/7 and if you subscribe to their announcements you'll get an email if there is an issue, and another when it's cleared, so they are on the ball.
As far as your specs go it's all ok...
- no live chat, you don't need it, tickets are very efficient
- as many databases as you want
- as many hosted domains as you want
- your own IP's if you use a Reseller account
- uptime stats at https://www.clook.info/helpdesk/index.php?/Network/Status
- no email blacklisting unless it comes from one of your own account
- PHPBB transfer, Clook may be able help transfer data, as with any server move it's wise to plan for some downtime
- Crons yes
- no host will support your coding but Clook have forums and you can still ask here, they may be able to assist if you think something should work but it doesn't, they're usually happy to enable misc PHP functions on request
- backups daily/weekly/monthly on a rolling cycle but as always do your own, there's never any guarantee that the latest backup will work