I completely endorse the ideas already mentioned: 1) connecting the metal can of the crystal to PCB ground, 2) improving the RF ground at the output (though more for output power reasons), 3) 0.001 uF at the DC input.
number 3) is, I feel, the most important. This will provide a low impedance to ground at 3-7 MHz and should help to shunt away the RF that is getting in.
Also, if there is not already a 0.001 uF across the two IC power pins VCC and GND, as close as possible to the IC power pins as possible, I recommend adding them somehow. One place to do this would be on the bottom side of the PCB, at the socket pins for VCC and GND. This is different than having a generic 1 uF across the IC power pins; the 0.001 uF is more likely to resonate better at 3-7 MHz and reduce the impedance at those frequencies across the ICs power pins. You want the impedance at those frequencies to be as low possible to inhibit RF from getting impressed upon the power rails of the ICs.
I recommend wrapping your DC feed wires on the left around a toroid core. Any type that you may have could be helpful: iron powder, ferrite, whatever, but ideally you want a ferrite RFI toroid. These are available from many sources. You can have the toroid outside the Altoids tin and you want to wrap the DC feed wires around the toroid core as many times as possible.
You could also use a 1:1 UNUN in line with coax of the beacon's RF output to inhibit the RF from your ham transmitter on the outside shield of the beacon's coax.
Good luck.