Is the presence of a magnetic field frame-dependent?

You are indeed correct about the frame-dependence of magnetic fields. The reason the point charge doesn't affect the compass is because the compass and the charge are both moving at the same speed, both being on the Earth, and therefore, the compass sees the charge as stationary. This means no magnetic field is produced.

As a side note, you hit upon an important realization: in order for electrodynamics to be consistent, you must adopt the same set of assumptions as in special relativity! In other words, special relativity is a necessary consequence of electrodynamics. Some books even derive the phenomenon of time dilation by considering the magnetic field experienced by a point charge moving parallel to a line charge.

This is a fantastic question, and you'd probably become famous for trying to solve it if it weren't for the fact that Einstein asked the same question over 100 years ago. The first sentence in his 1905 paper on Special Relativity says (translated into English, of course):

"It is known that Maxwell’s electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena."

He was referring to something along the lines of this: if a charged particle makes a magnetic field, then if I move at the same speed and in the same direction of this charge, I shouldn't see a magnetic field (he was actually considering moving a magnet through a metal loop, but it's essentially the same idea). This realization inspired him to essentially devise special relativity! HE found, in particular, that magnetism is actually an entirely relativistic effect. That is, magnetism only exists because of special relativity. If you walk past a stationary charge, you should see a magnetic field in your frame of reference.

This realization absolutely revolutionized the field of physics and indeed the world (Einstein would be considered a genius even if he never did another thing in his life -- but he did so much more). If you're interested in physics, you should pursue it. You ask the right kind of questions.

I hope this helps!

In the context of special relativity, the electric and magnetic fields are not distinct vector fields related through Maxwell's equations but are, rather, part of a more general higher rank (than a vector) electromagnetic tensor

The components of this tensor 'mix' when transforming between inertial reference frames so that both the electric and magnetic components are frame dependent.

Is the presence of a magnetic field frame-dependent?

In general, it is not possible to 'transform away' the magnetic field. Consider the simple case of two point charges in relative uniform motion. Since there is no inertial reference frame in which both charges are at rest, there is no inertial frame in which there is only an electric field.