There are three primary issues associated with using an auto transformer.
1) Some European 230v inverters neutral-ground to one side of their 230 vac output. This is absolute no-no for U.S. split phase, requiring the European inverter to be modified to disconnect the neutral-ground from one side of its 230vac.
2) Many folks incorrectly wire the 230 vac inverter to breaker box through dual 240v breakers and separately wire their auto transformer also through another 240v dual breaker in panel with auto transformer center tap neutral going to breaker panel neutral bus bar. If auto transformer breaker is opened while inverter is still supplying 230vac to panel you have a floating neutral situation in panel. 120v ac loads neutral will be free to float between L1 to L2 side dependent on how many 120vac loads are turned on each L1, L2 side. Good way to blow out a lot of 120 vac equipment.
Auto transformer must be connected directly to 240vac inverter output (with separate dual pole small breaker box between inverter and auto transformer) and L1, L2, and neutral from auto transformer brought into electrical panel with L1 and L2 fed into a single dual 240vac breaker. Center tap neutral can be directly connected to panel neutral bus bar.
3) When having connected grid pass through on split phase 240/120vac through inverter and neutral from grid is passed through to neutral output of inverter (auto-transformer center tap) any 120vac voltage imbalance from grid L1-N or L2-N will try to be corrected by auto-transformer. Voltage imbalance of 120vac sides may be caused by your neighbors 120 vac loads that share the same utility pole transformer. If you have a good quality grid pole transformer and grid connections it may not be a significant issue. You may have 2-3 amps of auto transformer load current caused by grid imbalance that will eat up a little bit of auto-transformer load capacity.
Other: These small auto-transformers being sold (approx 8" box size) are really only capable of maybe 2kVA to 2.5 kVA. This is normally okay as any of your 120vac loads that are on opposite L1-L2 side phase will not contribute any current through auto-transformer if the amount of 120 vac loads on each L1,L2 sides are similar. For example, if you have 3 kVA 120vac loads on L1 side and 2 kVA on 120 vac L2 side load, only 1 kVA balancing current is carried by auto-transformer. If you have a 6kVA inverter and 120vac load L1 side with 4 kVA and no 120 vac load on L2 side the auto-transformer will over heat since it must carry the 4 kVA balancing current. This is an unusual use case but it is possible. Be careful of carelessly wired kitchen with multiple 20 amp breaker outlets wired to same L1 or L2 phase side.
240 vac load contribute no loading on auto transformer. So any high current loads should be converted to 240vac if possible. A water pump is a good example of motor that can be rewired easily to run on 240vac instead of 120vac.