What do you think a "proper BA" would do?  Define that and then do it.  Voila, your company has a BA!

That's oversimplified and a bit trite, but my point is that this may be more within your control then you think.  You're asking for permission to do a job that you know needs to be done, but I doubt anyone would stop you from doing it if you carried on without a formal blessing (or job title).  I'm guessing that you are probably already performing tasks that could be considered BA tasks anyway.  When you're testing without proper requirement documents, do you ever have to go back to a developer or a stakeholder and ask how something is supposed to work or what result you should be expecting or where the data is supposed to be moving around?  That's BA work.  Think about how you can formalize that work, then start delivering it in that way.  Finally, measure the results of that work - did you uncover missed requirements, did you identify a business value that no one had considered, was your testing more productive?  Then you have a stronger argument for your boss.

Here is your first 'formal' BA task - develop a business case. In this case, you need to justify the reason for having business analysis activities standardized in your organization. Part or all of the solution may be to have a dedicated Business Analyst but there may be reasonable alternatives. Let's focus on defining the problem first.

You can start with identifying your performance measures: what activities are being performed right now that are impacted by haphazard business analysis? How much time do they take? In software development tasks such as function/feature specification, testing, and re-development due to missed or changing requirements are usually some activities that are impacted by BA processes.

Once you have these you can discuss how you think these areas can be improved with formal business analysis. Estimate how much time you think can be saved, either using some anecdotal evidence or some reasoned variance analysis. The goal is to demonstrate that by performing BA activities in a formal manner there is value to the organization (usually in time saved = cost avoidance/higher productivity). This presents the earning potential of your proposed change.

From there you need to look at options for improving BA activities. If you want to take on that role then you can just present this option, but you could look at other alternatives such as hiring a consultant to come in and develop a BA methodology that many team members will use, with no dedicated BA required. 

Then include the cost of investment for each option into the analysis. For the dedicated BA option if you feel you need BA training price out the cost of this training. Add in the cost of your time that you won't be doing your other assigned duties. You may also want to factor in going to BA conferences and/or other professional development time outside of the classroom. If you won't be performing QA anymore and/or there is a need to hire someone else to take over some or all of your QA duties include the cost of the new hire as well. Come up with a cost range for this and compare it to the anticipated benefits. If there's a net positive to the organization then you have a feasible business case.

Present your findings to your boss. If you have a solid business case it greatly increases the chances of getting what you want.

Good luck!