It takes time to fully charge a battery. 80% to 100% charged is no less than 3 hours no matter how powerful the charging source might be, so topping up via alternator, is not something accomplished in 15 minutes of idling. I know this is contrary to the public mindset that the alternator is some free magical instant battery charger, that the rules of physics no longer apply.
For best recharging performance, the thicker the copper the better, and the shorter the circuit, the better. So it is better to take power from alternator (+) stud rather than engine battery.
1, this bypasses the too thin cabling provided by the manufacturer
2, this allows the voltage regulator to see than higher field current is required to raise system voltage, and hopefully allows it to keep seeking 14.4v rather than prematurely dropping to 13.7v
3. Bypassing the engine battery, the circuit is usually significantly shorter.
A depleted healthy battery over adequate copper can max out an alternator. This causes it to generate a lot of heat. Temps over 220f are cumulatively damaging to an alternator.
An Alternator capable of high output at low rpms can cook if the vehicle's voltage regulator keeps demanding 14.x volts. Idling to recharge means lower alternator fan speeds and limited underhood air movement and is hardest on the alternator.
Much variation from vehicle to vehicle. My Dodge van alternator temperature skyrockets at max load at idle, but the same load at highway speeds has temperature down to cakewalk levels.
Yes you can make a jumper, across solenoid, or combine both cables to the same stud, or just feed 9+ volts to the trigger circuit on the solenoid.
Many circuits chosen to trigger a solenoid are live during engine cranking, meaning house battery is contributing current to starter motor. This is not desirable and will wear the Main contacts in the solenoid much faster. Find a circuit live only with engine running if possible, or put a lit manual switch to manually control the solenoid. many different ways to trigger the solenoid. If the Hvac blower motor turns off when engine cranking, this is a good trigger circuit for the solenoid.
AGMS like higher charge currents when depleted. An AGM used as an engine starting battery only in theory should never be depleted much, and as such could never accept huge charging currents, making this high amp recharge requirement of theirs in this application not worrysome.
If the engine battery is depleted and the house battery is depleted the batteries will share the current, but likely not equally. The AGM being lower resistance and on a shorter circuit, should pull more current, but there are of course variables which could make that untrue/ inaccurate.
I recommend using 2 digital voltmeters visible to the driver. 3 wire voltmeters with the third wire being a voltage sense wire hooked right to the battery (+). HOuse an engine.
These will allow one to see how long it takes house battery to catch up to engine battery, and to see if/when the solenoid contacts fuse together rendering no isolation with engine off, AND if the house battery is contributing to starter motor current undesirably.
It also reveals the wacky behavior of most vehicle voltage regulators. While we dwellers would love to see 14.4v allowed any time the battery is depleted below 100%, the vehicles VR often will allow something much less, increasing charging times slightly to greatly.
The one thing to always remember is that the batteries require time to reach full charge, time at absorption voltage, and this time is considerable, and reaching 100% regularly is the key to good battery life. reaching 90% only, will have the battery sulfate and lose capacity much much faster than if it reached 100%.
With 20 feet of wire one way to the house battery from alternator, I'd recommend 2awg or thicker cable. If you ground house battery to nearby frame, make sure the mating surfaces are shiny, the bolt threads also uncorroded, and use grease or liquid electrical tape. Adding a ground cable from frame nearby alternator to alternator (-) or alt mounting bolt is also wise. If not then the original engine battery to engine ground cable is carrying all the charging current to house battery as well.
Ideal ground path for house battery recharging would be another cable routed in parallel to the alternator (-) rather than using frame, but this is expensive and heavy and the percentage gained would only be realized when the house battery was depleted in the 50% range, and the vehicle was driving on the highway, where adequate rpms and underhood airflow could allow the alternator to max out and not overheat.
Idling unmoving to recharge is much harder on the alternator due to the alternator fan not moving fast enough and not enough underhood airflow.