Mortgage as an agency worker

theselfishaltruist

Hi Guys. I searched for some info on this, but couldn't find any posts discussing this issue specifically.
I am intending to buy my flat within the next year under right to buy (it is council). I am a trainee teacher. In September, I will be qualified (touch wood) and will be working  as a teacher. 
I have done the maths, and my mortgage payments will be around £600 (rising to 850 after 2 years). This is manageable on London teaching salary (which brings in around 1900 per month or 31k). 
Some banks consider the RTB discount as part of the deposit (I know RBS does), plus I have 20k in stocks. In total I am contributing 132k towards a flat worth 290k (although it might dip in value in the upcoming months, of course). I require a mortgage of 160k or thereabout and my LTV is around 55%.
I would prefer to work as an agency teacher as it is flexible.
My question is, will banks touch me as an agency worker doing supply work, or do I need to find a full-time job? If so, I will start looking for full time posts instead.
Just wondering if anyone had some experience or views on this. 
Would it be worth calling a bank specifically and discussing it? I'm aware I can get a broker, but tbh, I would rather do this myself as I believe I can do my own shopping around with banks, and therefore not have to spend money on one.
many thanks