dmu

That's what the Cassandra in me has been saying - not only students, but also (English-speaking) French Graduates who haven't been able to find work. I don't know how much a bar-tender earns per hour, but I imagine not enough to pay the rent and eat much more than pasta. If you find a bar and regular TELF work (and remember that you'll have to have a "status" to do TELF which involves paying côtisations), then it's feasible. But do bring savings to tide you over for at least 6 months while you're searching on the spot. "Barely scraping by" in France won't be a comfortable existence...
Is there a particular reason for choosing Toulouse? There may be more work opportunities in other cities with less competition on the employment front.

Wolvers

My French is conversational, I'm far from fluent but I can understand what's being said to me, read novels, watch TV/film without subtitles etc. From what I've seen the average monthly wage for a bartender is 1500 euros/month... I've had management roles in the past and I spent a month at bar school last year being trained up meticulously on cocktails and wines, so potentially I could find a better paid position down the line. But honestly, whilst that may not seem much I'm only 21 and I've spent the last three years living on £50-60/week, max; if all goes to plan I'll find steady teaching work a few months in, but I know a barman's salary is plenty to live on if you live within your means.

There's nothing explicitly tying me to Toulouse, other than that it's where I'm doing my TEFL training. But I've visited before and loved the city. Would I be better off looking into somewhere else?

So when you say savings for 6 months, I know the maximum I'm going to be able to take is £4,000. Rent shouldn't be more than 400 euros a month, and I'm accustomed to living on a strict budget; do you think that would be enough? I don't drive or have any regular outgoing expenses. Really appreciate all the advice, cheers.

dmu

Sorry, I understood casual "barman/maid" work in a café/pub. It's quite possible for a "bartender" to earn that sum, as a salaried employee in a hotel, for example, but there again, there'd be competition from French candidates, who have diplomas from Hotelier Schools, for any jobs going.
Come over for your TEFL training, with your accommodation organised for the duration and as much as you can bring over as savings, and see how it goes. You're only young once! Good luck!