Is it okay to connect a European 240V transformer to a US 240V outlet?
Yes, you can connect that to a US 220-240 V outlet. Just connect to the two hot lines and connect nothing to the neutral. However, there are normally only 30 to 60 amp 240 volt outlets available in US residences. They are dedicated to clothes dryer cooktop and oven use. To add a 240 volt circuit requires two adjacent 120 volt circuit breaker spaces in the distribution box. You will need a 15 or 20 amp double breaker whichever is the smallest available. You can connect additional 120 volt circuits to the same breaker, but they must not be circuits like bathroom and kitchen outlets that can not be mixed with outlets elsewhere.

It is not a good idea to connect to a higher current circuit because the higher rated circuit breaker might not prevent a fire if you have a short-circuit in the charger or in the charger cord. If you want to do that you should have smaller fuses near the outlet. I don't know if that is permitted by code.

You should use a US receptacle and plug rated for 240 volts and the fuse or circuit breaker current rating. You probably will not find a suitable converter for a 240 volt US receptacle.

There could be a difference in performance related to the actual voltage in the previous vs. present locations. However the transformer will be operating within the specified voltage and frequency range, so that is nothing to be concerned about. If the new voltage is higher, the charging current may be a little higher and the charging time a little quicker and the opposite if the voltage is lower. I assume an old charger may not have any voltage regulation. The operating temperature may be lower or higher.

A 15 or 20 amp 240 volt receptacle installed for a window air conditioner is an excellent choice for this application.