This will apparently be heavily scrutinized due to the time of year and the fact it would interfere with the current budget
Budgets are always scrutinized, but to play Devil's Advocate, I'd argue that your manager's departure has freed up plenty of planned expenses that could defray the cost of increasing your salary.
The two options you outlined are both maybes:
- Maybe we can get you a raise right now, or
- Maybe we can get you a better raise if you wait until EOY
Are these options substantially different in the short- to medium-term?
How much better if you wait until EOY? There's still a full Quarter of the year outstanding. A few percent maybe? Let's say you can get that $10K bump right now, or maybe a $12.5K bump in 3+ months. You'll earn an extra $2500 this year, and $10K more next year. At any point between now and 31/12/2020, you'll have earned cumulatively more money by taking the early, but smaller raise.
We can illustrate that with a few different potential increases starting on Jan 1 versus the smaller increase effective 1 October 2019:
Of course, as the illustration above shows, if you expect a much larger raise by waiting until January, the equation changes and your break-even point moves forward (the intersection with the black line).
Regardless of title/salary increase, your responsibilities are going to change, RIGHT NOW.
As someone who's survived several rounds of RIFs (reduction in force/headcount), I can give you my anecdotal experience:
When people leave, their responsibilities don't follow them. They're absorbed by other members of the team or department or whatever. If you're the only other member on that team, you're going to (obviously) incur the bulk of this. Some responsibilities may be tabled indefinitely or go away, but not all of them.
Unless the company has very good succession plans in place to reduce this sort of friction when employees/managers leave, you're going to start accumulating additional responsibilities that were previously your manager's. Right now. Some of these will be mundane, but some will not be. You're going to be doing 2 jobs, instead of 1. (OK, you'll be doing 1 < job < 2)
So you maybe don't have experience in all of those responsibilities, but who else is going to do them, if you don't? And what is the cost to the business if those things don't get done at all? It's probably significant.
I would push strongly for the title + wage change, effective ASAP. Balance this with your understanding of a skill shortfall and try to get a plan of action in place to bring you up to speed as quick as possible; this may involve shadowing someone in another similar role in a different department, or attending internal or external training (on the company's dime), or online tutorials, etc.
Also note that in a year's time you'll ideally be on par, or if not, you'll have a year's experience and better prospects for seeking other opportunities.