According to various fact sheets on the Tenants Union website the landlord is required to provide the accommodation in a reasonable state of cleanliness and repair and you are required to maintain the cleanliness and leave the unit in the same condition as you found it.
With that in mind, it sounds to me like your rental agent is an ignorant bully (one of the very few, hey?) who needs a quick lesson on the workings of the Residential Tenancies Act.
I would write them a letter saying that the state of the oven is not your concern as you don’t use it and the landlord (or their agent) thought it was OK to hand it over in that condition.
However, if they want it cleaned you will allow reasonable access for a cleaner paid by them to do it.
In the meantime, do you think the rent increase is excessive? If so, there are also factsheets at the above website on how to challenge that, too.
And on the subject of ovens, if you do decide on a professional clean, preferably paid for by the landlord, go the whole hog and get the ‘Oven Fairy’ in. Seriously, check THIS WEBSITE out – they’re not cheap but your oven or barbecue will be as good as new when they’re done.