The reason a landlord asks the questions about rental increases is because they know and believe the premises are below market.
In your position, the premises should be advertised for rental with an anticipated vacancy date of two (2) months. Prospective tenants generally begin the search a few months before intended move. Since the current tenants do not respond well to rent increases, advise them that "You are aware of the discomfort and inconvenience of increased rent and would like to provide the opportunity to find a new apartment in advance of the rent increase."
Hopefully, this will place both parties in a position for reasonably discussion about the move or the rent increase.
ALWAYS maintain the rent at the fair market value!
I think this is a good idea. Let the current tenants know that you need a decision about if they are renewing at the higher rate or leaving within 60 days of lease end, otherwise you're not renewing. If they don't renew, start advertising at the 60 day mark. The better prospects do start looking a couple of months ahead of an anticipated move date.
My guess is that the tenants make a big deal out of the increase because they either feel the property is not worth more or they can't afford it.