There's a shortfall in production.
Current capacity of 6000 pa. Demand of 8200 pa. Very roughly, a shortfall 2200 pa.
You say:
Each employee added can produce an additional 20 units per month and is paid $1000 per month.
Which is the same as saying each additional employee can produce an additional 12x20 units pa. (=240).
So you could cover that 2200 shortfall by employing an additional 9.166 employees.
You try employing .166 of an employee! Actually it's possible with timing, BUT you also say:
Changes in production level cost $100 per unit due to hiring, line changeover costs, and so forth. This is an astonishing figure, and a bit woolly. To what units do you add the $100? All of them?! Just those produced in one month at changeover? Whatever, it looks very expensive. The message being: avoid changes in production level.
In the attached there's a cell B16 where you can plug in the additional employees and some dead simple formulae so you can see inventory levels over the year:
Here I've used 9. Interestingly, the remaining inventory at the end of the year is only 40 less than it was at the beginning of the year (if you plugged in 9.1666 you'd get the same inventory at both those points, of course). It's also interesting that the minimum inventory is 80 which is about 10% of max monthly demand - which might be considered a safe level to cover unexpected demand.
Now if you plug in smaller numbers (in cell B16, you'll start seeing shortfalls in some months. I'll leave it to you to do some what-if calculations to see the costs of making up those shortfalls with overtime and/or subcontracting.
Here it is with 8 additional employees:
You could cover shortfalls with additional overtime if you knew that the higher demand in September/October was seasonal; you can get 10% of 660 (=66) more units per month on overtime alone, so if you were to get max overtime in Jul/Aug/Sept/Oct you might cover the shortfalls in Oct/Nov without any other method of increased output.
Overtime looks like it might be more expensive than subcontracting since overtime costs an extra $25 per unit where subcontracting is only $15 extra per unit.