What prevents the water switching out over and over is that your opponent has to send a pokemon into a mrmime psychic.
Look at it this way, these are the pokemon that can switch into a MrMime psychic >50% of the time and beat it (All calculations from this point on desregard ches, with the exception of the move slash):
Arcanine, Flareon, Ninetales, Pidgeot, Rapidash
Note: Rapidash is only listed because of Firespin.. If it were to switch into a Thunderwave it will lose. Also, if Pidgeot switched into a Thunderwave or Thunderbolt it also will lose. For Ninetales and Arcanine, I haven't calculated exactly, but I am pretty certain switching into a thunderwave would drop their odds down to below 50%.. Which leaves flareon as the only safe switch in.
The following is a list of pokemon that can switch into MrMime and status it, but will most likely die before being able to finish it off:
Electabuzz, Electrode, Magneton, Mr. Mime, Parasect, Porygon, Tangela, Wigglytuff, Abra, Clefairy, Dragonair, Drowzee, Exeggcute
Note: Venomoth can achieve this also, but it needs to attack first, not be ched on the switch in and actually hit with whichever status move it was using, so I didnt bother listing it. Also Abra, Clefairy, Dragonair and Drowzee are not going to be met in uu battles..
And for fun, Pokemon that beat MrMime one on one:
Arcanine, Electabuzz, Fearow (with D-E + Hyperbeam), Flareon, Golduck, Magneton, Mr. Mime, Ninetales, Omastar (with hydropump), Pidgeot (with D-E + Hyperbeam), Rapidash, Raticate (with Superfang + Hyperbeam), Scyther, Vaporeon (hydropump), Wigglytuff.
Which granted is quite a few, but most of these are very unreliable, Golduck for instance needs to survive three turns without being ched, and then needs an R value of > 226 when it attacks.. Also MrMime could thunderbolt as Golducks statups then switch to a faster water with a physical move. If Golduck attacks before Amnesiaing twice it will lose to MrMime.
Also, if you are going to allow MrMime on the basis of its awful defense, why are you banning Dugtrio? It is faster, but is an easier countered attacker and is even more fragile than MrMime is.
Have a nice day.