Roger P's absolutely right.
Nothing like a bit of adverse publicity to concentrate the mind of some complacent manager.
Ideally, find the nearest councillor to you who is in opposition to the ruling group. It's Lib Dem minority control; you have two independents and a Tory nearby, so that's a good start.
There is also
http://whatdotheyknow.com for making FOI requests. This site is really brilliant.
You might, for example, enquire who runs the tip, how big the contract is financially, how long it has to run and how many staff are employed there. You might also ask to see the description of the service the contractor is being asked to provide, and any constraints on them concerning what they may (or may not) do with material deposited at the site.
You might also ask about any other contracts that contractor has with the council - how big they are and when they were awarded, etc. Crucially, you could also ask how many formal complaints have been received about that tip, broken down by month received, and how many were resolved to the satisfaction of the complainant.
Let your imagination roam free!
Don't expect decent answers on the first attempt. In my experience (and I have had some success using whatdotheyknow), they can be devious little *&*#$^*!. They are, however legally required to respond, and if they attermpt to fob you off or redact important stuff you can appeal (and should).
If you do use whatdotheyknow.com, do NOT give them the slightest hint of why you are asking. If they are behaving this badly in one place, chances are the whole operation is at best very badly run, at worst - well you can speculate yourself. Sharp questions with no obvious point of focus frighten the daylights out of them.
The above forms the basis for a campaign, but it also serves to alert middle managers that the natives are restless. You might, just, see an improvement without needing to go any further. It's worth talking to one of your councillors too, but you might gain more traction initially going the DIY route.
I have uncovered alarming abuse of public funds through doing similar, but so far have been unable to get much done about it, as it is a small department, part of a huge administration itself renown for waste. My conclusion was that in the great scheme of things it was simply too small a problem.
But that issue although infuriating, didn't directly affect me, whereas yours does you.
If we as citizens aren't prepared to hold the batsards to account, nobody else will.
E.