Whether you need a second degree or not depends on what you mean by "current level of experience with physics". There are many people who enter PhD programs in astronomy or astrophysics with degrees other than a BS or BSc. But these people would have taken the same courses (as electives) as those with science degrees.
For an astronomy or astrophysics PhD, most people enter with an astronomy or physics undergraduate degree. An astronomy undergraduate degree is very similar to a physics degree. I think if you want to be eligible for PhD programs in astrophysics or astronomy, you should look up the physics degree requirements for your school and see if you took enough of these math and physics courses. Note that physics degrees tend to have a lot of required math and physics courses!
You say that you already graduated. If you have most of your physics requirements met (i.e. less than one year's worth of courses left), then perhaps you would be able to take these remaining courses as an unclassified student at your current school? If you do this and do the Physics GRE, you should be eligible for most programs.
However, if you only have a few physics courses completed, I think a second Bachelors degree is a good idea. Some schools will allow you to complete a second degree in just 2-3 years because they will allow you to waive all your breadth requirements/electives since you have a BMus from the same school already.
Also--physics is not the only route to astronomy/astrophysics. I don't know about the other routes as much because I didn't go that way, but I've met some people who enter astronomy with Chemistry, Math, Computer Science, or electrical engineering degrees. Of course, these people would be entering specific subfields of astronomy or astrophysics related to their background. And they would all have some basic physics and math courses (they would still need to be able to complete graduate level astronomy/astrophysics courses).