The clippers probably won't mind a modified sine wave inverter, but they may be noisier on that than on a pure sine wave one.
Conversion via 12 V is one of the easiest ways of doing it because both the 230 to 12 V converter and the 12 V to 110 V converter are readily available.
Efficiency is usually defined as power out divided by power in, and that won't be good if you convert via 12 V, but I don't know if that is important. The cost of electricity will be low on the list of priorities for a set of clippers, as they are quite low power and they probably won't run for long. The efficiency will be poor because there are multiple stages in the conversion. It is probably something like this:-
12 V power supply:-
Rectify 230 V rms 50 Hz mains to give DC at 350 V
Switch 350 V DC on and off at around 30 kHz
Transform 350 V 30 kHz to around 12 V (peak) 30 kHz
Rectify 12 V 30 kHz to 12 V DC
110 V inverter:-
Switch 12 V DC on and off at around 30 kHz
Transform 12 V 30 kHz to about 160 V (peak)
Rectify 160 V (peak) to give 160 V DC
Switch 160 V (peak) on and off at 60 Hz to give 110 V rms 60 Hz
As you can see, that is a lot of stages, with power lost at each stage.
I don't know what you mean about accuracy. The frequency and voltage produced by a cheap inverter won't be very precise, but why would you care for clippers.
I think that clippers are a rather different device to many, if they are anything like the electric razor my father had. The cutters actually move at the frequency of the mains. The voltage causes current to flow, which causes acceleration of the cutters, so speed builds up, and the cutters move, and that is stopped by the mains voltage reversing and the whole lot happens in the opposite direction, with the process repeating 60 times a second.
If you run from 50 Hz, there is more time for the current to build up so more current, more time for the acceleration, with a larger current, so much more speed, and then more time for the speed to give distance. It is likely that the noise is the cutter hitting the end stops hard. This could be made worse by a transformer that gives out 120 V or so with a full load, and your clippers are just a small load, and the transformer voltage is higher
It is common for electrical devices like transformers and motors to work on a lower voltage with 50 Hz than with 60 Hz. In the case of the clippers, you may need to look at a much less, maybe 110*(50/60)^3, or about 65 V if they are to work at 50 Hz. If you do that, the cutters will run more slowly, but you will probably only notice that the pitch of the sound they make is lower.
You could try something like this:-
https://uk.farnell.com/multicomp/mcta250-35/transformer-toroidal-2-x-35v-250va/dp/9530800 but that is quite expensive for a test which might not work.