Regarding the op, personally, I wouldn’t say no to someone who is a shtickel zionist.
I think there are two basic issues:
1. What does it mean that he is a shtickel zionistic? Are there any practical ramifications, and if so, what are they? For example, if the guy would send his kids to the same schools as your daughter would and has the same type of Rav and wouldn’t dream of sending his son to the army, respects the (Chareidi) Gedolim and the Chareidim, and the only practical ramification is that he has a bar-b-que on Yom HaAtzmaut and/or says Hallel without a bracha, I don’t see what the big deal is.
2. #1 may have to be qualified depending on your daughter’s background/upbringing. When I wrote #2, I was thinking of myself. For me, coming from the background I am coming from and having the friends, social setting, and mentality that I have, it might not necessarily be such a big deal. My age also may make a difference.
On the other hand, if you are talking about a girl who is coming from a very insulated, super-Israeli-Chareidi background, it might be a bigger deal.
But even so, if she is happy and they have similar goals and values, and he has good middos, and #1 holds true, I wouldn’t recommend objecting.