If there are no 240V loads (seems likely) and if your inverter package has an appropriate output breaker to protect itself from overload (since the panel is clearly going to be able to draw far more than the inverter supplies) you should be fine, but you're almost certainly going to need a custom-made cable, not a pre-purchased adapter.
The L14-30 has hot1, hot2, neutral and ground. A 5-15 has hot, neutral, and ground. The expected voltage between hot1 and hot 2 is 240 volts. If there are only 120V loads, you can connect both hots on the L14-30 to the single hot on the 5-15 and it will work, up to the point that the inverter overloads (not too likely if your loads are as described, but then someone plugs in a coffeemaker or hairdryer.)
As the breakers are currently arranged, feeding only one of the two hots would leave half your circuits without power.
Most of your more serious off-grid inverters will happily supply 240VAC, and don't come with a receptacle at all, as they expect to be hard-wired. If you are going to connect a tiny inverter and keep the loads tiny, it should work with an adapter made as described.
I strongly suspect that you can do a lot better (cost wise) with a built system than the pre-packaged one.
Ok you wrote a book. Proposing all manner of third rate hackery. And what does it boil down to? You want to get 5000W out of your 5000W generator. Quick question.
What is 240 x 21 ?
By my math, it's 5040. There's your 5000W. You do get it out of the big NEMA L14-20 connector.
I have no idea where you got 41A. I'm pretty sure you made that up, probably by dividing 5000 by 120. I seriously doubt it was on the generator spec. There's a way if you really really want that, but as you get educated, you will realize you do not.
What is it you're missing? The odd idiom of North American 2-pole service. I don't blame you for not getting it... It's weird.
Your house is served by +120V, neutral (0V), and -120V. I just described an instant in time, they're AC so they will reverse position 120 times a second. The poles are called L1 and L2 and the middle is Neutral.
240V loads grab L1 and L2. 120V loads grab either pole and neutral. Which pole they grab is nearly random and that's the idea, to make them average out so loads are balanced.
For you, with 21A on each pole, balancing is a big deal. You'll have a problem if you put 30A of load on one pole. So you'll need to get into the gory details of what is on which pole, and manage accordingly.
Step 1: Control MWBCs so they don't kill you
I don't recommend rearranging things on a panel because you can break a type of wiring called a multi-wire branch circuit. Find an electrician and tell him to do exactly this:
find every multi-wire branch circuit in my home, and make sure both its hot wires are served from the same 2-pole breaker.
Step 2: get rid of double-stuff breakers
If your panel is stuffed, and has lots of breakers that have 2 breakers in 1 space, those will drive you absolutely bat crazy. ack... You know what, to heck with all that.
Let's just get you a new subpanel with the appropriate interlocks, and move the loads you want the generator to power into this new subpanel. Make this subpanel quite large (at least 20 space) realizing you'll use 4 spaces just for the interlock.
In a perfect world, your new panel will have ammeters which will tell you how close to 21A each pole is getting. Even better get one of those new fangled whole house monitoring systems. Ask a new question on how to get one to work in a generator interlocked panel.
Step 3: rearrange your loads in the panel
Now finally, it's time to learn the gory details of how poles are assigned in a panel. Read my posting here. Your panel may differ, but probably not by much.
Move your loads into the new panel, and consciously and carefully balance the loads. For instance if your table saw is on L1, put your dust collector on L2. Stuff like that.
Best Answer
Each switch has 3 positions, as labeled on the right hand side of the panel: GEN OFF and LINE. In their current position, the normal electrical feed is going to each circuit. If you move the switch to 'GEN' it will then take power from the generator. OFF lets you kill the circuit entirely (connects to neither line nor generator).
If the panel was properly installed there should be no need to kill the breaker at the main panel: Flipping the switch on this panel disconnects the given circuit from the main panel.