The monotone convergence theorem handles infinities gracefully, which can only be done for functions that are positive (or otherwise reasonably controlled from below). In particular, nowhere does it assume that $f_n$, $f$, or the integrals, are finite. This seems to be beyond the scope of Beppo Levi, so I'm not sure that fixing this issue alone is considerably easier than proving everything from scratch. But let me try.
Depending on your version of definitions, it may or may not be trivial that for a positive function $f$ the following special case of monotone convergence holds:
$$\intop_E f dm = \lim_{C \to +\infty, E_n \uparrow E} \intop_{E_n} \min(f, C) dm$$
where $E_n$ are sets of finite measure that approximate $E$ (I assume $\sigma$-finiteness; if it fails then we should restrict to $\{f > 0\}$; if it fails even there then monotone convergence holds almost trivially with both sides infinite).
Now in order to make use of Beppo Levi we should make the limit finite. I would do that by replacing $f_n$ by $f_{n,C,k} := (f_n \wedge C) \mathsf{1}[E_k]$ and $f$ by $f_{C,k} := (f \wedge C) \mathsf{1}[E_k]$ for some fixed $k$. Now we can safely apply Beppo Levi to the successive differences $f_{n+1,C,k} - f_{n,C,k}$ to obtain
$$\intop f_{C,k} dm = \lim_{n \to \infty} \intop f_{n,C,k} dm$$
Now take $C$ and $k$ to $\infty$ and interchange the limits. You can always do this with monotonely increasing limits (this is equivalent to rearrangement of terms in positive series, or Fubini on $\mathbb{N} \times \mathbb{N}$, or whatever you prefer). On the other hand, monotone convergence itself is about rearrangement or Fubini on $E \times \mathbb{N}$, so I'm not even sure you would view the things that I rely on as more basic than those that you prove...
Looks good, here is a shorter argument with the same idea:
The sequence of numbers $\int f_n \rightarrow \int f$ if and only if for each subsequence $\int f_{n_k}$ there exists a further subsequence $\int f_{n_{k_l}} \rightarrow \int f$.
Let $\int f_{n_k}$ be given, $f_{n_k}\rightarrow f$ in measure, there exists a further subsequence $f_{n_{k_l}} \rightarrow f $ a.e. and $\{f_{n_{k_l}}\}$ is UI, apply Vitali we have
$\int f_{n_{k_l}} \rightarrow \int f.$
Best Answer
There are proofs of the monotone and bounded convergence theorems for Riemann integrable functions that do not use measure theory, going back to Arzelà in 1885, at least for the case where $E=[a,b]\subset\mathbb R$. For the reason t.b. indicated in a comment, you have to assume that the limit function is Riemann integrable. A reference is W.A.J. Luxemburg's "Arzelà's Dominated Convergence Theorem for the Riemann Integral," accessible through JSTOR. If you don't have access to JSTOR, the same proofs are given in Kaczor and Nowak's Problems in mathematical analysis (which cites Luxemburg's article as the source).
In the spirit of a comment by Dylan Moreland, I'll mention that I found the article by Googling
"monotone convergence" "riemann integrable", which brings up many other apparently helpful sources.