You will be a lot happier if you learn about two important business concepts that may apply in your case:
- productivity is not the only goal
- local vs global maxima
Whenever a developer says "I would be more productive if..." and yet they aren't allowed to do whatever that second half of the sentence is, they say it's "politics" and want to know how to force management to let them. And to be fair, sometimes it is politics. The boss came up the ranks using VB and he likes it so everything these days is in VB.NET. The boss golfs with a guy from Oracle so everything is Oracle no matter what it costs. If you're in that situation, give up. Learn to like using those tools or get a new job. But do you know the boss' career background or who he golfs with? There are other explanations.
When you work alone, all that matters is your own productivity. Use whatever tools you like, work whatever hours you like, adopt whatever processes you like. Sometimes you might use the free tool because you can't afford the paid one, but to be honest it's pretty easy to get free copies of expensive tools, especially for a startup. (Eg BizSpark gives startup ISVs all the Microsoft software they need, for an unlimited number of developers, free.) But when you work with others, things like productivty are still important, but they're no longer the only thing. Predictability - everyone's ability to know what is going to happen when - suddenly matters a lot. For example, say you are planning to code a small (4 hour) change Monday, to be tested by your people Tuesday, the client rep Wednesday, and go live Thursday. But Monday morning you have a headache and feel crummy. You realize it will take you 8 hours to get it done that day. You would be more productive putting it off a day. Problem is, even if the others can find just as productive ways to spend their days, you've disrupted their plans. You might find the IT guy refuses to deploy on Fridays so he has to wait till Monday. And that might change a line in your manager's spreadsheet from "deployed week 33" to "deployed week 34" - all because you wanted to be more productive Monday. This is true at a small scale and also at large. Being able to stick to plans has value, and sometimes it's more valuable than productivity.
Other things have more value than productivity too, like manageability and transparency. So if you want to use a free text editor to write your code, but it doesn't integrate into the work item tracking system, it's not ok for you to personally decide to opt out of the work item tracking system in the name of your own productivity.
Then there is the matter of optimizing your own productivity (local maximum) or optimizing the productivity of the team as a whole (global maximum.) To use an example on a small scale, if you want to work midnight to 8am, that might make you more productive, but it means other people who have questions for you have to email them, then wait till the morning to read your answer. The search for your local maximum has moved the team as a whole away from the global maximum. Or if you create docs locally using Word, but the team is all using Google Docs, there will be an impedance mismatch where you don't upload things or download things and others assume yu have. Same problem. This also happens at a larger scale you may be unaware of. That database you all find so hard to work with may be just what some other team needs to do the reporting. Or that service bus that's adding a layer of indirection and making you crazy may be vital for the customer's communications with another partner. There may be a solid business reason why your team operating at a lower-than-possible productivity is in fact best for the business as a whole, just as there is a solid reason why you operating at a lower-than-possible productivity is best for the team as a whole.
Of course, it's possible that nobody ever gave any thought to another way of doing it. Perhaps your proof of concept will be met with opened arms, and everyone will swing over to the new way immediately. It can't hurt to show your boss. But if it's rejected, understand that it may be because of golf or personal prejudices, or it may be for good and solid reasons that you just don't happen to know.