Many large companies sell their product, many councils use it. I phone businesses constantly trying to get them to choose a uk product over a chinese import - that I can match on price - and better , but they aren't interested in us, only the competitor - because they are a bigger company.
Step 1, analyse. What is really the problem? The root cause.
Step 2, come up with solution options.
Step 3, pick one
Step 4, implement
Step 4, monitor adjust and control, then loop back to Step 1.
Trying things like price increases, reductions, more advertising, changing your product etc... no offense but doesn't it all seem a bit headless-chicken? Its jumping straight to steps 2 and 3 - and solving the wrong problem.
Based on what you've said above, the problem may actually have little to do with features, quality or price... just buyer confidence.
Because many other councils use the competitor product, and because its a big supplier, this gives a lazy buyer more confidence in that product than in yours. The Halo Effect. You need to drill down and find out what the buyers are really thinking... and that will be the clue to why they are not buying your product.
For example, being a bigger company doesn't in itself make your competitor more appealing. What is it about their size that makes them more appealing than you? Are the council buyers afraid that you may disappear in a few years, so they would have no after-sales support for the product so would end up with a worse product in the long term? If so, address that somehow.
Maybe its nothing to do with confidence. But you will only ever find out if you start understanding your customers rather than trying to push your product.