There were two kinds of PP4. The original had almost x2 capacity of the PP3 as it was square cross-section, if the layer cells filled the box.
Later they made it round.
An alkaline PP3 will not fit in a sensibly made cylindrical PP4 as the width is too close to diameter. But it will fit in a square cross section PP4.
Only 5 AAA cells fit in the round PP4, but you can fit 6 x AAA (without a holder, so a custom holder or soldering needed) in the square version.
A PP3 will fit in the square one too. But best is to use the Alkaline PP3 and open it. (not all Alkaline makes are welded AAAA cells, which is what is needed, some are held in place by case.
6LR61 = 6 off AAAA size Alkaline cells = IEC LR61 cells
6F22 = 6 off Zinc Layer cells = IEC F22 cells
(Though not all Alkaline PP3 are exactly the same size!)
Compared are PP3 Zinc layer battery (6F23), AAAA (or similar) cells from Alkaline PP3 (6LR61), an AAA cell and an N or Lady cell (1.5V, similar size 3V lithium and 12V alkaline exist!). The N cell is too long to stack two in length and fatter than AAA, so no help at all.
Best is to open the Alkaline PP3 and cut the connector insulator in half, and cut the negative connector strip in half. Add a wire and double sided sticky foam pad to put the connector on the bottom.
It may then fit in the round case, but not if the wall is decent.
It fits easily in the squared case:
Compared to the Battery Valve boxes I have been making, this is like a battery for a dolls house!
Here it is full size! (300dpi, print so when folded it is 50mm high, each side just over 25mm, just under 1").
Fold the inner most tabs at about 47mm to 48mm spacing depending on card, as you have 3 layers of card on top and bottom!
Either print on 160mm card direct or 90gm paper and "Pritt" stick or spray mount to cereal box card.
Regard this is a prototype as I don't have a radio that uses PP4.
Pity the square version was replaced by round and that we only have the PP3, it would be nice to have 9V alkaline that was essentially 6 x AAA cells rather than the 6LR61 Alkaline as it's only 60% improvement on layer PP3 and only long life in smoke detectors.