Refinance or use own funds and talk to the bank?

And regarding "should I tell the bank that I am putting in more equity/would they 'reward' that" -> for the current financing, that probably doesn't make a difference at first. If what you put in now increases the value, it may be that your house is ultimately worth more than in the bank's original appraisal. However, they won't give you any great bonuses for your ongoing financing because of that. It can be relevant if you want to refinance the house again and in that context the loan-to-value ratio, etc. is reviewed. Or also when it comes to follow-up financing in 10 years. Things like that, where the actual house value becomes relevant again.

if you have enough equity, then pay with that.

with the "direct bank" the refinancing currently probably costs the current interest rate 2.5-3%) + 1% surcharge, so 3.5-4.0%

if you have an investment where you earn at least 5%/6% or more each year (from which we deduct capital gains tax etc.) then 3.5%/4.0% remain.

therefore, the decision should be easy

If you have not borrowed money at a favorable rate from the beginning, a follow-up financing is always more expensive

if you have enough equity, then pay with that.

with the "direct bank" the additional financing would currently probably cost the current interest rate 2.5-3%) + 1% surcharge, so 3.5-4.0%

so if you have an investment where you earn at least 5%/6% or more per year (from that we deduct capital gains tax etc.) then 3.5%/4.0% remain.

therefore, the decision should be easy

So if I have €30,000 and can invest it at 5% return, I do not pay capital gains tax because I can first use up €1,600 tax allowance. In that case, I would refinance as long as the interest rate is below 5%.
If I am afraid of the stock market and the money would just sit in the account anyway, then I would put it into the house.