The answer to your question under those specific circumstances is yes. They can ask you to leave and if you do not then you will be arrested for trespass. I had a very similar incident with my son a couple years ago, non gun related, but in the end I did find out that they have the authority to remove anyone from a park that is owned by the municipality that emplys them. I did not re-look up the code but if anyone wants to throw the flag I am more than happy to provide it.
I disagree with it very much but the law is pretty black and white about it.
You know, that almost rings a bell of truth. However, looking at
RCW 9A.52.090In any prosecution under RCW 9A.52.070 and 9A.52.080, it is a defense that:
(2) The premises were at the time open to members of the public and the actor complied with all lawful conditions imposed on access to or remaining in the premises; or
Thus that leaves out any Criminal trespass charge. I don't believe that a LEO can make "can't open carry a firearm" a "lawful condition imposed on remaining in the premises".
I'm sure there must be some obscure law about having to follow a command given by a LEO, I just can't find it.