As you suggested, most hand held radio chargers are designed to charge the battery, not charge the battery and run the radio. Doing both might work with some charger/radio combinations, others it may not. What will likely happen, is that over time the rechargeable battery will be damaged from constant charging. Once your rechargeable battery goes, you'll have to buy a new one, which can be expensive.
What I'd suggest is to purchase a permanent mount Marine VHF radio and a 12 volt power supply. This will prevent the battery issues and it will give you much better audio. Rather than trying to use a hand held with a small speaker, or using an external speaker plugged into a jack designed for headphones, get the mobile. The increased audio output and better speaker will make for much more pleasant listening. Add an external speaker and it'll sound even better.
Here is one on sale for $97.00 Only slightly more than the cost of the above mentioned handheld.
http://www.defender.com/product.jsp?path=-1|344|2028693|2028953&id=1701917
For the additional cost you'll prevent having to replace batteries every few years, and you'll enjoy better audio.
Of course, you'll need to figure in the cost of a 12 volt power supply. For listening only, I'd suggest something with at least 2 amps output. This is getting beyond what most wall-warts can do. Finding a small regulated 12 volt power supply will take a bit more work. Here is a very basic 12 volt 4 amp power supply:
PYRAMID PS2K 10A AC to 12V DC Adapter : Power supplies | RadioShack.com
Of course with a radio like that, you'll need an antenna. Actually, getting an antenna mounted outside the house, will improve reception. Finding a VHF antenna won't be hard. You could get yourself one of the fiberglass white VHF antennas used on boats, or you can use just about any antenna designed for VHF.
Either way, hand held or permanent mount, it should address the inter mod issues, and will have much better sensitivity than a scanner.