Whether there's actually a difference between AVERAGE and PEP power depends on the transmission mode used.
If your Daiwa meter doesn't register anything at all in the PEP position, I suspect one of two things:
1. It is a "powered" meter, requiring connection to an external DC power source in order to work in the PEP mode. (Does it have a DC power socket on it? Possibly one labeled "12Vdc in" or something? If so, you need an external power supply or the PEP function will be "dead.")
2. Or, there's simply something wrong with it (defective).
A "true PEP" meter MUST have a source of DC power applied to it, as active, powered circuitry is required to make a true PEP power measurement and that power cannot be taken from the RF applied. A battery, or a power supply must be used.
On FM, AVE and PEP are identical.
On AM, PEP is normally 4x AVE carrier power level, for 100% modulation.
On RTTY, AVE and PEP are identical.
On CW, PEP is the only thing that counts, and AVErage power will vary depending upon what you're sending. If you send a long string of "di-dahs" (like the letter "A") with no pause between the letters, e.g., didahdidahdidahdidahdidahdidah, etc, the AVErage power output should be about 50% of the PEP.
On SSB, there theoretically isn't any AVErage power because if you stop talking, the power output drops to zero. So, the AVErage power while taking a pause is zero. On the other hand, if you continuously speak, and speak quickly, there will be some AVErage power indicated, usually about 25% of the PEP power.
PEP = peak envelope power. Commonly expressed as the ratio of the square of the peak RF voltage across a 50 Ohm load. For example, if you measure 100Vpk across a 50 Ohm dummy load, the PEP power that represents is 200 Watts [(100*100)/50 = 200].