Battery showing flat

The simple answer is as your are seeing 13.2 with the charger on, it is unlikely. The 13.2 is a bit low but a dead battery could easily account for that. 

The next question is the battery dead in not being able to accept a charge, it is certainly flat and if left like that will be dead.

I would try charging it with one of my many stand alone chargers, but things are "hinting" a bit towards a dead battery.

Float is what the charger does when the battery is as full as the charger can get it.
Now that is very different to the battery being as full as it once could have been; its just as full as it can now be. A natural ageing/use process is a battery progressively loses holding capacity till for the user that capacity, or lack of it becomes intolerable.

To see if you have a battery issue or an in van charger issue these crude layman’s checks will give some clear pointers;

With fully recharged battery reconnected to van charger, with no real loads in use and EHU on, what is the voltage?
If 13.4 and above then the charger is trying to do something so is likely to be okay. If below this then it is not or the battery’s issues are lugging it down.
Leave off the EHU for 6 or more hours, again with no demands being taken from the battery.
If then the voltage is 12.7 things are healthy, the battery has retained its charge.
Still off the EHU, then put on a load, such as the heater blower or similar and leave for 4 hours then switch it off and remeasure the voltage. If that is much below 12.6 volts then the battery’s capacity is not much good; some might find it adequate rather than buy a new battery but it is not a lot of good any more.

Armed with these values we are in a better position , though not infallible one, to know where your issue is.

I have seen in the past many reasons for batteries not showing the correct voltage so befor you go shelling out money and if you have a multimeter do a bit of checking....

Measure with meter on the 12v (may be 20v) the positive post to the clamp around the post. if there is no indication of voltage then all is well there. then do the same with the negitive post.

Find it you can find the negitive return via the chassis for the circuits you suspect, it will almost certainly under the chassis near the house harness, then with a long cable that will reach the negitive post measure the resistance between the two, it should be 0.000 ohm's.

It that is OK then start suspecting you battery.

If you ask a battery dealer to measure it beware of digital reader's, some are not acurate but ask for a "High rate discharge tester" that will tell the truth.

No matter what the outcome is from now on always smear a little vasalene over the posts, it stops the lead from corroding and building up a resistance.