The problem is that the output of the inverter is
PROBABLY floating. I stress the probably because they vary in design, but that's a logical way to build it. Therefore, unlike the mains where (on a TN system as almost all of us have) neutral is to a greater or lesser extent connected to earth - it's not in your inverter. IMO the boiler design is rubbish - in fact I'd go as far as saying it's incompetent as there is really zero need for that limitation.
Provided the inverter output is in fact floating, then it would be possible to connect N&E together and it would run the boiler. However, I would caution against that for the simple reason that it creates a significant potential hazard of making the inverter case live in some usage/fault combinations. For example, someone borrows the inverter for another task and plugs in some item of Class I equipment - that equipment develops a L to local earth fault which will pull the L of the inverter output to local earth. The inverter is still happily putting out 240V and so now it's L is earthed, and it's N is flapping around at 240V. Because you've connected N&E inside it, the metal case is also flapping around at 240V waiting for someone to touch it at the same time as something locally earthed - perhaps the case of the Class I equipment. Said someone is now completing a 240V circuit
The issue is that while you may understand that borrowing it and not securely earthing the earth terminal is dangerous - you can't guarantee that everyone else will understand.
That's why I suggested adding (for example) a shrouded 4mm socket to the neutral connection of the inverter - and using a separate lead to the boiler/plumbing earth. With no plug in, the inverter is floating again - and the socket is fairly finger proof. So if someone borrows the inverter, they may wonder what the extra connection is for, but there won't be the hazard that a fixed N-E link would create.
Let me tell you about something that these days when I know was was going on makes me cringe. Maaaany years ago when I were a young lad, I used to work on a farm. We had this great long extension lead that would get from one of the few sockets (I only remember there being three altogether) to pretty well anywhere we needed it. The fact that it was rubber, and many cracks had been covered in tape is not relevant - other than showing it had been around for a while. The socket was one of the old (waay back when the brand was synonymous with quality) MK surface sockets screwed on a piece of wood - and sticking out of the top was a coil of single core cable with a crocodile clip on the end. The great long length of cable was, it turned out, two core.
At this point, I think a few people reading this will immediately recognise what this arrangement was supposed to be. Just think about this - it's a two core cable with a 13A plug on one end, and a 13A socket on the other end, not normally considered a safe arrangement.
But none of us had any idea - sometimes the croc clip wasn't used at all; sometimes it was clipped the machine the lead was powering (such as a metal framed conveyor for hay bales); occasionally it was clipped to a water pipe - all screwed galv steel pipe back then; eventually it was declared a bloomin' nuisance and chopped off.
I'm sure that when the lead was made, instructions would have been given for how to use it safely. But over time, and with different people about, that information was lost. And as a result we were often using completely unearthed equipment (these were the days when a Black and Decker drill had a metal body and a 3 core flex), that was not always well maintained, and not always kept and used in dry conditions. Hopefully the danger in that is obvious.
I think many will have recognised that this arrangement was completely safe
if used properly - and indeed would have been to avoid a significant hazard from using a regular 3 core extension lead. The croc clip was intended to be connected to a good local earth - such as the galv steel water pipes - and thus provide an earth that's local to the user. Using a 3 core extension lead means a) transferring an earth from one part of the premises to another at a risk of there being dangerous differences in potential during fault (and even some non-fault) conditions, and b) having a significant resistance to earth that could cause dangerous differences in potential during fault conditions (e.g. a fault in a hand tool). In extreme, b) could cause the earthed body of the drill to be at half the mains voltage until a fuse blew - which could conceivably be a while with a very long extension cable. And on a farm, you have livestock that are far more susceptible to voltage differences than we are (normally).
And that is the sort of thing that results in rules prohibiting certain things if used by unskilled people. If someone is trained in understanding the risks and how to mitigate them, then things can be allowed that you cannot trust "some random person" to use/manage safely.
But back then I knew nowt about this sort of stuff, we didn't have RCDs and stuff like that, fuses were mostly rewirable with the replacement wire based on one or both of a) what's to hand, and b) what is thick enough not to blow again. Except for the welder, one of
those Pickhill oil filled jobs - that had a piece of bolt in the plug. But somehow we survived - although I recall the wiring setting on fire one day when we were pressure washing a building out, water got in the cracked rubber and it started spitting and smoking, but it was OK after drying it out and putting some tape round it
I do sometimes wonder how much of that survival was down to wearing rubber wellies most of the time
