The basic cell voltage is too different between Nickel Metal-hydride and Lithium-Ion batteries. A single-cell NiMH battery cell is similar to other AA size rechargeable batteries at 1.2 volts per cell while the Lithium-Ion batteries are around 3.6 volts per cell. As you stated, a fully charged Lithium-Ion cell is about 4.2 volts which is below what the radio needs to operate. 2 cells would be 7.2 volts rated and 8.4 volts fully charged. So a 2-cell Lithium battery is too high a voltage for the radio.
There's no good way to run a radio designed for 4.8 to 6.0 battery volts on a Lithium-Ion or LiPo battery pack. The voltages are just too different.
As for charging - no way. The Lithium chemistry requires accurate constant-current, switching over to constant-voltage charging to prevent heat, puffing and possible fire or explosion of the lithium battery(s). You cannot charge a multicell lithium battery pack in a series configuration without per-cell battery management (which the Home-Patrol, of course, lacks).
If I were you I'd stay with quality, low self-discharge NiMH batteries and charge them externally only. Set the charge time on the Home-Patrol to either 0 hours and charge them in an external smart-charger or set for only a low charge time of 1-2 hours to keep them topped up in the car. If you use the scanner on batteries, reset the charge time to an appropriate value for the amount you've discharged the batteries, then reset to 1-2 hours afterwards (or go the external smart-charger route).
As has been discussed here many times before, the HomePatrol and other similar devices designed to charge NiMH batteries internally do not have the proper circuitry to perform 'smart' per-cell charging. You will always risk overheating and scanner damage if you set the charge time wrong (too long) for the estimated amount you need to recharge NiMH batteries in a series configuration.
Unlike older NiCad batteries, NiMH doesn't have a linear voltage charge/discharge curve. Also unlike NiCads, NiMH has an exothermic charge response which generates a lot of excess heat. Without per-cell voltage and temperature monitoring, even NiMH cells are at risk for overcharging and radio damage. You can forget about Lithium batteries altogether - they're just too different and require even more specialized charging.