Get an ICB report, costs around 6Eur, will give you an idea of what you look like financially. based on what you have said alone, it won't nessesarily discount you from any bank, but if your other finances are in risk category, your reconable income isn't great (i.e. circumstances change like another kid etc) then even though can comfortably pay mortgage now, computer may say no on those and additional factors. LTV is good and mortgage isn't crazy high so they are in your favour.
you will need to check but choose the order of what you go for, like KBC, UB, BOI, PTSB, EBS, HAVEN, AIB etc - take into account things like cashback/solicitor contribution, also if want to just move and stay with that bank, or if plan to move again in years time or so to avail of additional cashback - all these come into play so you may not do your first move with the best option, you want to end up with the bank with the best rate (or thereabouts) and good history of treating customers fairly
FI and Dilosk don't have the best rates, so personally I would leave till last to try them (and pepper, would need to check but they might be worse again)