If I book an Anytime Day Single from BSK -> STP via Any Permitted Route, will I get the cross-London transfer?
I believe there is no such thing as a ticket from Basingstoke to St Pancras. Your ticket will either be issued to "London Terminals", with no †, or to zone 1 (for an extra charge) in which case it would be valid to any underground station in zone 1. (This won't have a † either, because it's for travel
to London, not crossing London. But it'll work the tube barriers.)
That being said, assuming you're connecting onto a Eurostar, there's a special destination available for this purpose called "London International (CIV)" (some systems have slightly different names). The station code is LNE. These fares are often have less onerous peak restrictions than normal tickets, and include the tube transfer to St Pancras. (They may also offer additional passenger rights, but this isn't particularly clear. In practice, in most cases connections are honored even on normal tickets.) From Basingstoke, the anytime single fares to LNE are £32.40 via Woking, or £37.50 via any permitted route, less with a railcard. If you're traveling back via the same route, returns are available and offer better value.
You can buy these fares from brfares, Trainsplit, or an especially competent ticket office. They are valid only in combination with a Eurostar ticket.