Here's the thing. You're thinking this through from the point of a company having to report their money and thinking, "Bruce gets all his toys from Wayne Tech so they have to hide it in some way." This isn't the case. He uses Wayne Tech R&D, but nothing off the books in that way. And anything he does use like that it's replicated on his scale, not using Wayne Tech, but however he does it; personal workshop, personal manufacturing, hiring out people, etc. There is no "hiding things on the books". Lucious doesn't do anything with the financial in regards to what you're asking. What he does is fund allocation and help give Bruce access to and apprised of that information without Bruce having to look into it directly or allocate funds for projects directly and look suspicious.
This brings us back to the question at hand, "the how does he fund it all?" It's an insane answer, but there are 2 sources. Any finances he does that has to do with Wayne Enterprises is Petty Cash. He does use this for various things, mainly on the fly when needed things that turn up right that second things. The other source is, as I said, insane. It's his liquid assets that he has. Obviously he doesn't keep all his cash on hand, but he assuredly has millions on hand in Wayne Manor.
I know you're thinking that it's impossible, because noone is that rich. Even the richest men on Earth don't have that kind of money to keep that much on hand. I'd agree that it's ludicrious to assert that this is where the money comes from (well that which can't be written off in some other way for example, hiring construction crews to build supports and such in caves under a house isn't all that strange when the entire area is on a fault line with sink holes and caves running through it...) but in the case of Bruce Wayne, it is true and Bruce Wayne really is "that rich". How rich you ask? Bruce Wayne is the primary owner of the biggest most successful conglomerates in his world, many of his companies being worth multi-billions and he's the primary stake holder of those too. That puts him as richer than any person on real Earth you might be thinking of with that alone. He is also finances a good number of all the business in Gotham (it's something like 1 in 4 to 10. It's been a while since I saw the number). So he's got lots of capital coming in from all of that, but then he also has his Mother's family personal wealth (The Kanes. I think they went Bankrupt though I forget) and Father's family's personal wealth which was separate from the corporate endeavors. The Wayne Fortune has a bunch of literal gold bars like fort knox as a part of it, but ignoring trying to find out what else it has in it there is only 1 thing that is really important from it, and that is Bruce Wayne literally owns 1/3 of Gotham... before getting into any of that other ownership that his Company or those businesses own or anything like that. All the land that is Gotham, at least 1/3 of it is owned by Bruce Wayne. So what does this mean? Gotham is one of the biggest cities in the world and is supposed to be equivelant to a New York. Theoretically Gotham's land should be worth more than New York's and New York's land is, if I remember right, the most valuable in the world.
In other words Bruce Wayne is literally worth Trillions of dollars without even discussing his corporate holdings, probably in the multi hundreds, if not tens of trillions, not Billions, Trillions. He has that much cash to just have on hand.
This fact, along with the Playboy, risk taking, crazy buying, persona hides all the costs he needs. And yes "Batman" is a multi-million dollar expense to set-up, but not as expensive as people think, because they things that generally get trashed aren't as expensive as you might think, or don't get trashed in the way that makes them expensive...
Another thing that hides it, which is weird, is that he is constantly upgrading and manufacturing things that are expensive like the different Batwings and Batmobiles. It looks strange for you to have a sudden expense that just happens to match the cost of repairing something that Batman just lost. It doesn't look strange if your costs are constant.
So... Lots of untraceable money + understanding how things look to people who investigate this stuff + being able to explain a lot of things due to the circumstance of Gotham
