Going in asking for costs at the start might not be a good look since the Tribunals are intended to generally operate with each party looking after their own costs. It does not cost much initially to apply for an order. Be clear about what you are asking for. Most likely that would be for an order that your owner must comply with the breach notices.
If the owner continues to offend in the same way, they would no longer be just ignoring a breach notice, they would instead be failing to comply with a Tribunal order, which is a more serious offence.
If you are lucky, just going to the initial mediation or directions hearing, which are relatively informal, may be enough to make it clear you are serious and for the owner to decide to behave better. You can ask for ‘consent orders’ at that point. If the owners says ‘I won’t do it again’, you can ask for a consent order, that is an order that both parties agree to, to give that promise the extra weight of an order. If they do do it again, you are immediately in a stronger position. Tribunals don’t like it when their orders are ignored.
While it is daunting at first if you have no experience, the Tribunals are accustomed to dealing with ordinary people who have no legal training. The main thing is to be very clear about what order you want. It has to be very concrete – that some person does or does not do some particular thing in some stated time.