The service dept was saying that since there is no coolant, there is no reason to take apart the water seal? It's just oil. If coolant was leaking, then there would be additional problems.
Since the shop/dealer's shown some kind lack of knowledge about the Yamaha FZ6 water pump, I suggest you should get the bike home, order the parts and do it yourself!
My words: follow the services manual steps by steps,,,carefully - and you are talking with me, a guy just messed up a set of full water pump rebuild parts but managed to repair it after all!, I'm here to help! hehehe (I have a sample broken water pump at home,,,when I get home I will take some pictures to show you how it's easy to rebuild your water pump!)
The only way to remove the oil seal is to remove the water seal. Once the water seal is removed, it can't be reused because it's required to use brute force (lol - sound like computer attack worm) to punch that water seal out = damage it in the end!
You can however save the bearing

, leave it alone, push the oil seal out from the bearing side.
The new oil seal must be pushed in from the water seal side - why? = the oil seal needs to push all the way until it's contacted the surface of bearing EVENLY! That's why the oil can block the oil to get into the weep hole or prevent the oil leak through weep hole. If oil's leaked through weep hole, it's oil seal failure! (oil seal surface with spring face to oil source or bearing - Thanks FI!)
Then it's required to use the new water seal - try to re-use the water seal = risk to get water into the engine

...and you know it's not an engine any more once coolant gets into it!
There is no better way to install the new oil seal without damaging the water seal!
Good luck!