I would say there isn't one straightforward answer, but I try to give a few hints/considerations and my personal assessment.
There are three main types of inductors (simplified):
- unshielded (coil exposed to surrounding air, magnetic field not shielded)
- semi-shielded (coil usually covered with mixture of ferromagnetic material and epoxy resin)
- shielded (coil fully encapsulated with ferromagnetic material)
Starting from the first type the shielding effect obviously increases accordingly. And also, the better the shielding, the better the magnetic flux surrounding the coil is held in place.
From a magnetic flux point of view it is better to have no copper traces/planes around the inductor. Thus, the magnetic field lines that form around the inductor are not significantly disturbed and the data sheet values can be better maintained. However, for shielded (semi-shielded) inductors the magnetic field is far (a bit) more concentrated around the coil and adding a GND copper plane underneath it will have way (a bit) less impact on the inductors properties.
From an EMI point of view I would say it is generally better to have a GND copper plane underneath it - this plane acts as a shield and protects nearby conductors (or inner layers) from picking up radiation. But keep in mind that the magnetic field gets disturbed and inductors properties are degraded (i.e. reduced inductance, ...). However, for shielded (semi-shielded) inductors this degradation is again way (a bit) less impacted.
Conclusion: If you are digging deep it gets complex but shielded type inductors are generally a little less critical to handle, which doesn't mean you don't have to think about what you're doing, but if it were my design, I'd try it with the GND vias mentioned underneath the inductor and have a good time working together with your colleagues from the mechanical engineering department ;).
I also see no risk with vias placed below regarding solder bridges with this type of inductor.
As a side note: Don't forget about the dot indicator (winding start) you'll find on many inductors. It tells you were the winding starts and you should connect this side close to the switching node (high dU/dt >> long traces would favor the antenna effect...)
PCB design for low-EMI DC/DC converters
The Behavior of Electro-Magnetic Radiation of Power
Inductors in Power Management