Quick summary of martingale method: Bet X, if you lose next time bet 2 times X. You will cover all your losses on the next win.
So the martingale method seems quite good on paper but doesn't work in practice due to consecutive losses blowing your budget.
For the sake of assumption, lets say we bet on 1000 coin flips and want to make a profit.
Does anyone know of any methods or theories that work in a similar fashion but can handle a limited budget?
No, assuming a finite budget, there is no betting system that can give you a positive expected profit from playing a fair game. This is a consequence of the optional stopping theorem.
If we let $X_n$ be your wealth after playing $n$ times, $\Delta_n$ be your bet on the $(n+1)$-st game (which can only depend on the outcomes of the first $n$ games), and $S_n = 1$ if you win the $n$-th game and $S_n = -1$ if you lose the $n$-th game, then $$X_{n+1} = X_n + \Delta_n S_{n+1}.$$ This is a martingale, i.e. $\mathbb{E}[X_{n+1}|\mathcal F_n] = X_n$, so if we play the game any fixed number $N$ times we have $\mathbb{E}[X_N] = X_0$, i.e. we aren't going to win or lose on average. This is true regardless of whether or not we have a limited budget.
If we don't fix the number of games to play in advance, but instead have a stopping rule that depends on the outcomes of the games (e.g. stopping when we run out of our budget, or once we hit some goal), then we do need the fact that we have a limited budget. The limited budget tells us $X_n \ge K$ for some fixed constant $K$, so if we say that $\tau < \infty$ is the stopping time, then we can use Fatou's lemma to conclude $\mathbb{E}[X_\tau] \le \mathbb{E}[X_0]$.