If I had £3k to spend on a startup, I'd try to only spend £2k or not much more, and have money left for all the unexpected expenses etc. It doesn't matter how well you do your homework, you never think of everything!
Start with a cheapo van & roof rack
Basic trad kit (Bucket, 14" wagtail, scrims, blades, BOAB, pouch, trad pole, spare rubbers, small squeegee)
Ladders (short lightweight triple extension would be my choice)
DIY van mounted WFP system with a CLX-27 pole
DIY static setup with pumped RO (Vyair.com are cheap)
DIY website
20,000 cheap flyers
Logo'd polo shirts
Basic vehicle sign-writing
PL insurance
Its quite a big ask for a small budget, so keep an eye on the buy & sell section here & on ebay for bargains!