So as a suggestion you'd want something like this:
GPS08-15U PS2 Connector Mini size Waterproof GPS Mouse (U-Blox 6 UBX-G6010-ST) | eBay
it claims to be waterproof (might not be IPx7 but you know keeping anything 100% dry on a SIB is a challenge so even in a water proof compartment this may be better!) Its more expensive than some others - but some of the others I quickly looked at were wrong baud rate I think. (Not sure which SH Radio you have but I think SH like 4800 baud with GLL, RMB, GGA, GNS sentences.) This one defaults to 4800 and GGA is in its list. GGA contains time of fix, position and some quality info about the fix. All you need!
GLL and GNS are essentially the same info in a different format. RMB has bearings to a waypoint. The mouse doesn't know where you are headed so can't send that info. Not sure the SH radio would display it if it did?
(Most of the explanation is here: NMEA-0183 messages: Overview
Power with something like this:
Waterproof DC/DC Converter 12V Step down to 5V 3A 15W Power Supply Module | eBay
Quite a few wires in there (I'd whip the PS-2 plug off personally). Then I'd connect from the mouse:
Pin 1 (GND) to Yellow Wire on the power supply AND the NMEA - INPUT on the radio (?green)
Pin 2 (5V) to red next to the Yellow on the power supply
Pin 3, & 5 not connected
Pin 4 may not be needed but if in doubt to NMEA + OUTPUT on the radio (?grey)
Pin 6 to the NMEA + INPUT on the radio (?Brown)
Red and black on the power supply to the 12V battery (red via a fuse +/- switch)
Test... ...will take a few mins for first fix. This unit has a LiIon battery which means it stores the almanac from the sats for up to 7 days. If first fix is after 7 days expect slow first fix, otherwise 30seconds roughly then every second.
I'd use a sticky pad to fix the GPS in place with top surface up. If your compartment is under the seat not sure if the seat will shield signal. adding a person to seat will...