Since I've been away for a couple of days, this is more or less echoing what was stated above. But so what I'm reading now is the following, kind of laid out in steps.
1) You have a 1000 watt inverter that you'd like to hook up to your house bank.
2) You have a bicycle battery pack that you'd like to charge, and it takes 230 watts and typically charges via AC.
3) 230 watts is ~2 amps AC or ~20 amps DC.
4) You have a ~200 amp lead-acid battery bank.
So, a couple of things I'd want to know would be these:
1) What is the actual voltage the battery pack takes to charge (yes, it plugs into AC now, but that doesn't mean it actually charges via 110v - there may be some built in transformer or etc.). Is there any sort of "car" "airline" "cigarette" adapter sold for it? There may not be, because most cigarette plugs are only rated for around 15 amps. But... worth a check (maybe the actual charge rate would be less without the AC brick or whatever).
2) How long does the pack take to charge?
There is the Peukert effect which basically says that even though you could run a load of .5 amps for 100 hours (to use half your bank), that doesn't mean you could run a load of 50 amps for 2 hours. Basically, a larger load draw makes the batteries take it on the chin much more than a longer duration smaller load. (Lithiums don't have this issue, but most of us probably won't have those for another round or two of battery replacement.). Not that this directly applies as you are talking "only" 20 amps and only when you are running the engine (or plugged in), but just mentioning it in passing.
On the face of it, this looks potentially doable to me. Of course make sure you have good fusing, etc. Actually, you may not be able to fuse the line coming back to your house bank from the start battery given the smaller wire (not that that's a new situation with the setup under consideration).
I've been thinking about what I think might be a fiendishly clever solution to something that has been annoying me. Namely, carrying around a 180# generator in my best outside storage locker when I use it about ten minutes per month, on average. But I DO have a few tasks I like to use it for. These are short duration, infrequent tasks. For example: 30 seconds of heat gun to shrink tubing over electrical connections, 45 seconds of running a trim router, a minute or two of Multimaster tool, a few minutes of vacuuming.
I haven't executed this on my Chinook yet, but I did run a test on a friend's rig, who has the V-10 (same alternator), same heavy wire back to house bank, same AGM battery bank, same battery monitors, and a 2,000 watt inverter (I'd be doing 1,000 or 1,500 watts, but no matter as I had the same loads running either way). What I did was turn on the inverter, start the engine, combine the house and start banks (we both have manual switches for this, but same concept as 7622). Then I ran each of my tools and watched what happened. What I wanted to know was, would the alternator rise to meet much or all of the load?
What I observed was that it did. Watching the Victron amp counter meter, there was a range from none to very little draw on the house bank. Most of the power was coming from the alternator. In other words, nothing my house bank couldn't easily handle. (We're talking about draws of around 600 watts by the tools.) Not that I would use my engine for an "hours long" generator without driving - that's not good for an engine that wasn't designed to be doing this. But my scenarios are all in the 30 second to 2 minute range (especially after I get rid of the darned carpeting!). Not that this is exactly what you are proposing, but I thought I'd mention it as tangential to the subject at hand.
So you are talking about a smaller draw, over a longer period of time, and you won't be running an unloaded engine (because you'll be driving).
I don't know how your inverter hooks up, but I'd encourage a proper installation. Hard wired, good fusing on ample wires with not too much voltage drop, etc. And it would be nice to go to a wire size from the start battery that could actually be fused. In my experience, 2AWG is the bare minimum, and 1/0 is better. Reason I say that is that my buddy with the same engine/alternator/starter (2003 V-10), didn't want to run much larger wire, so decided to run 2AWG. With the purely recommended fuse size for that wire, a self-jump would blow the fuse. But invoking the "if you have to you can fuse a bit higher" rule (it's worded a bit differently of course), he was able to make it work. IIRC, it was a 175 that was too small, and a 200 that worked, but I'd have to check to be sure as this was a couple of years ago). He also has more voltage drop than he would with larger wise, but it's much better than the completely unfused 6AWG that he had originally.
I went with 1/0 (considered 2/0, which would not be bad either) and don't have any problem fusing within guidelines. I started with a 225 amp fuse, and kept a 250 in reserve in case that wasn't enough (1/0 can handle either no problem). 225 has been fine though.
Just for the record, the ampacity tables show that 4AWG (Chinook stock), if it's 105ยบ C rated wire (most Chinook wire is, and I think the 4AWG was too), and since it's in an engine space (heat), it's rated for up to 136 amps (if not bundled). Even fusing 50% over that (ouch) is only just 200 amps. I believe that's why it's not fused in stock form. The 2AWG my friend used is rated to 178, which is why we first tried the 175 amp fuse. But it was too small to take the starter's load. So we went up to 200, which is still in the acceptable range, if not perfect (less than 20% over). The 1/0 can take a 242 amp fuse as rated, and I have had no problems even with a 225 amp fuse.
BTW, while not a ten minute job, running the new cable wasn't all that bad. This is also a perfect time to add a 7622 and place it somewhere besides the fenderwell (I know you already have a 7622), if so desired. Buying the larger cable is slightly ouchy, but coming from boating RV's always seem like a bargain

(and I hear that airplanes make boats look cheap, so there's always that). One note is that in order to complete the circuit, you also need to have the negative wire. Chinook uses the frame (chassis) for that. BUT, the Ford start battery has a short little wire from the start battery negative to the chassis. So that will also want to be upgraded. That was actually harder than running the big long wire aft to the house bank! And there is the negative wire from the house bank to the chassis ground (that was pretty easy).
Okay, that was probably more than you wanted to read. But, cloudy morning, lounging around vs. starting tasks
As always, I'm not an electrician (not even close) - so if anyone sees any holes in my logic, please do mention them. I can always learn something and also don't want to steer anyone wrong.