Start by going back to the original dealer and see what he has to say. Keep your cool when doing so. The other dealer may be authorized, but you didn't buy it there, and he isn't going to go out of his way to help. I personally don't think the second company did a very good job of diagnosis. I don't know how low battery voltage and seized engine can both happen at one time and be related. At this point do not try to unsieze the engine until the dealer has had a chance to look at it.
Sounds to me too the second dealer diagnosis may be questionable - if the mower was running ok before you changed the oil, then subsequent to your oil change didn't start & only "clicked", it sounds unlikely the engine is seized (combined with the low voltage it seems more plausable a faulty starter/solenoid or battery is causing the drama)
However, assuming worse case scenario & the engine is seized at only the 5 hour mark I'd be pushing back damn hard for warranty/consequential damage based on poor pre-delivery &/or supply of faulty goods:-
For your mower to seize after only 5 hours either the machine was not pre-delivered with the correct oil level, or it has consumed &/or leaked oil at an extraordinary rate indicating a faulty engine.
As Rivets advise keep your cool (but be firm with the dealer on your expectations, & also call the manufacturer direct) you should not be out of pocket for this...
Good luck & let us know how you get on.
The way I see it you have added a second problem now. You are now going to have to get two dealers to agree to what the cause of the problem is. This is not going to be easy, as they will each have a different reason to say it is your fault, as you have already heard. I cannot say who is responsible, because I cannot see the parts. The cause of a siezure could be a variety of things. To list a few, but not limited to; faulty engine part, (like rings not seating) improper dealer setup, (not enough oil, loose drain seal) operator error, (not checking oil). If I would try to say who should foot the bill, i would do a disservice to everyone using this forum. The good techs on this forum never take sides in this type of problem, that's not why we are here.
I just had a second thought. (Only had one cup of coffee this morning) I would contact the engine manufacturer directly and explain your problem to them directly. Let them know that you probably have put two dealers against each other and that is not what you want to do. Admit that you just want your unit fixed and may not have gone about it the best way in the heat of the moment. Don't forget to mention this forum and the advise you received here. Don't put the blame on anyone yet. Many times working WITH the manufacturers rep. Speeds things put and makes everyone look good and feel better.
Charge the battery