Can we write to an internal register of WDT to extend the time of counting? Let's say we want to implement bit-banging.
A WDT always has a "clear" mechanism, in fact this is how they work - your code needs to clear it. Normally this is done through some kind of path that guarantees the µC works normally, for example a timer interrupt. If something truly bad happens like interrupts end up disabled this timer interrupt will never happen, the WDT doesn't get cleared, and so will trigger an hard reset when you're out of time.
Also be aware that once you run the WDT you can no longer debug your code - it will stop on a breakpoint, but as soon as you try to step or continue the WDT will issue a reset and the debugger will get really confused. So the WDT is typically something you add as an ultimate reliability mechanism to make sure your system can't wedge hard once everything already works like it should. But you could still have major software bugs, the WDT only prevents hard seize-ups. And you don't want too much complicated code in the way of clearing the WDT, or you may find you have bugs in this code and get spurious nuisance resets that are REALLY hard to track down.