Joren already gave you a perfect answer for what to do, so this is more of a comment...
"Put another PCB antenna on the other side of the board, overlapping the first antenna" - This may seem like an okay idea, but is actually a bad idea, for several reasons.
A PCB trace antenna almost always requires nothing conductive near it - other than a ground plane in a specific place and of a specific size. Putting copper on the other side (whether it's an antenna or anything else) will detune it and/or cause lots of reflections, that's why most PCB antennas have a keepout that requires no copper on any layer, and also anywhere within typically a distance equal to their size (or preferably much further). They're detuned by dielectrics too, but less so.
The two antennas will resonate together. If you prefer, since both are on the same band, the inactive one will receive a really strong signal from the active one - and then probably retransmit it. This will probably not work out the way you want (ie neither will work properly), but also probably will exceed the spec for whatever is connected to the inactive antenna (as most Bluetooth chips don't really expect a 0dBm-level signal coming into the RF port). The least bad thing that can happen is harmonic generation which is out of spec.
Because of 2, the detuning is likely to be much worse from a nearby antenna than from (for example) simply any random nearby conductive object. I've ran Bluetooth antennas very close to a metal plate - it basically works, although the range is not good. I think running an antenna close to a second antenna could (for example) give you a bunch of directional gain (and correspondingly, directions with very low gain) due to intereference patterns. Whatever it does, you'd have to simulate or try it, but it can probably be summed up as "nothing good" :)
Now, since I do want to encourage you to get hands on experience with RF, what I would suggest is you take two PCBs you already have which have trace antennas, stack them the way you want, tie the grounds together (hopefully all along the edges or better) and then hook up a network analyzer to one of the antennas. The N1201SA is a decent inexpensive device of this type that shows you what's going on with your antenna (maybe not as accurately as a Keysight or Anritsu, but good-enough). You can cut a micro coax cable and solder the end to the feed point of the antenna and the shield to ground, the other end with SMA connector goes to the analyzer. Then, let us know what you see so that we can all learn :)