Metrics to prove or disprove the need for additional DBA's?
Demonstrating ROI on hiring additional DBAs is hard since DBAs don't generate revenue by themselves. They do however allow the organization to function and generate revenue, just like any other supporting staff. A lack of staff in any department makes the organization function less optimal and lose money or lose opportunity. So there is a cost from lack of investment rather than a return on investment.

If you feel you need additional DBAs you must know what tasks you are unable to perform because of lack of time/skill, if it's lack of skill  (such as the SSAS stuff you mentioned) that can be trained, if it's lack of time it could be temporary or structural. I've been in situations where you just know that your life would be a lot easier if you could just automate this and that or set up some alerting but just never get around to because of too much reactive or repetitive work and general fire fighting.

Once you have determined why your team isn't functioning as well as it should it's time to document it.

First of all, document what you do now, why that takes up all your time and make clear up front to your manager the existing problems that need solving are absolutely none of your team's fault. Make sure you don't give the impression that the problem would be solved by hiring 'better' DBAs.
Make an excel sheet showing your workload for the coming weeks if you can, charts impress, and it can show your workload in a single slide.

Once you are clear on that, document all additional tasks and everything that needs improving, but especially document what the business risks and possible costs are, and the savings could be. Also document how much man-hours you think would be needed to make those improvements and perform those tasks.

Things that go on the investment side of the equation are


Salary of the newly hired DBA
Price of training


Things that go on the savings side of the equation are


Projects not finished on time
Preventing downtime that makes the business lose money
Preventing legal risk
Preventing data loss
Consulting fees could be reduced
Cost of burning out employees because of too much work


Once you have compiled the list as complete as you can it should be clear what the best price/reward option is


Getting additional training
Getting additional DBAs
Hiring a temporary consultant to assist in automating stuff, monitoring stuff, performing a migration so that the existing team has more time
Changing the priorities put on you by the business so you get time to get things in place that will allow you to work more efficiently later on