Do you have just the one capacitor? I mean if it was wired wrong and you had two capacitors in series around the power rails then the most each would ever get to is half the battery voltage.
Another thought is that once the voltage gets high enough a relay is trying to click on but the relay plus the powered load is higher current draw than passing through your resistor because the ohm value of it is too high. I don't know this to be true, and it would seem unusual, but "could" happen.
Can you link a spec sheet and instructions or even better a schematic of the attached circuitry integral to the capacitor-product?
I can't speak about this specific capacitor product, but in general you should be able to measure the voltage going in on the +/- power rails and it should be very nearly the same as the battery voltage, IF measured before rather than after the resistor where the voltage will be lower.
It is the ohm value of the resistor that determines how fast it charges so using the same ohm value but moving from a 1/4W resistor to a 1W resistor only decreases the amount it heats up (same amount of heat created but lower thermal density). You may NEED at least a 1W resistor to have it not heat up too much, but "capacitors" in general would charge fine at a slower rate with a higher ohm value and thus not need as high a wattage rating. The circuit they've added to it is the wild card.
I would contact Makamichi or an authorized distributor and ask them if the added-on circuitry has any charging complexity requirement.
What ohm value resistor are you using? If you were starting from zero and managed to get it up to 6V, you are definitely charging it, and the same resistance value should continue to charge it but at a progressively slower and slower rate as the capacitor gets near the battery voltage, but at only 6V, it is nowhere near that and you should keep seeing voltage rise unless there is an interaction with the attached circuit eating the limited amount of current going to it, as mentioned in my 2nd paragraph above.