Thanks for your input guys, I'm thinking of just buying the 6V adaptor
Don't forget that 4 x 1.2V rechargeable batteries, would be about 4.8V, and very slowly drop as they are used. Also a 4.5V adapter would tend to be a bit higher voltage in practice, depending on the load. 4.5V is very close to 4.8V, so surely it is designed to cope with that voltage ?
Or has the instruction manual BANNED rechargeables, because of the slightly lower voltage ?
tl;dr
If it seems to be working fine with the 4.5V adapter, and the battery low warning light is off (not showing), you should be fine.
Battery voltages usually drop, and drop while they get used up, so it should be designed to work at voltages below the 6V. Usually a battery warning symbol is designed to show, when the voltage is too low.
As regards some posters worries about it overheating (which I think is very unlikely to be an issue). That is usually when you give it TOO HIGH a voltage. 4.5V should be designed for, since the batteries, as they wear out, can easily reach that voltage and lower. So if overheating was going to be a problem, it would anyway, with battery use.