You can't have it both ways. .
If your company (or does such as BASC still claim to be a charity ?) is paying for the service to e.g. allow you to work from home (lucky you) then they can, and must, set the rules. and make hard decisions. Failing to prepare is planning for failure ISTR,
Simplistically there is a whitelist, and a black list. Yes that is how they are named, and convened to compare notes. Either in person, or online, mostly on the dark web.. Just as there are white hat and black hat groups.
If BASC have (unsurprisingly) got onto a blacklist, well I doubt that they are sufficiently clued up to rectify that.
Preferably over a secure specific VPN, with monitoring of everything that you do, or attempt to do. Records to be kept. Possibly a little incompetent. If they don't have experienced skilled IT staff who understand the rules. In short supply it seems, according to the recruiters that I have chatted with Or so it would seem.
In what way might perusing SD or many other websites that have no business nor political influencing value be deemed appropriate ? Do it just through your own home arrangement.
Try going into a public library (if you can find one open) and try to access SD Or other such sites. Or anything slightly slightly controversial . It is quite scary how little is now available. How did that happen ?