Just hand-waving the numbers a bit, it looks like from a strict financial perspective your market looks like it favors renting over buying.
I would also not buy a house "for future children," but that's just me. Extra space in the meantime will cost a lot. If you do go through with this, definitely rent out the remaining rooms to make some of that cash back and put the space to work.
prices are expected to increase 6-7% this year.
Do not hang your hat on "expected"s. It could also go down, and in such a high-priced market, it could go down by a lot. Especially if you're only putting 5% down, could you stomach a real estate market drop of 10% after you buy? 20%? You don't want to get stuck. Maybe hope for 5%, plan for 0%, and have a plan B for down to -20%.
It also sounds like you have very little spare liquid capital. If you move in, the market tanks, you lose your job, and you find out the house has foundation issues (or insert other scary-expensive-problem X), what do you do? Can the two of you live off just your girlfriend's income (or just yours) if it came to that, while accounting for a house?
I'm not trying to scare you; I just want you think of all the permutations and possibilities. If you walk through the worst-case scenario and survive that, you can be much much more comfortable making your decisions.