Your picture of the U-100 Syringe (a insulin syringe) is correct for 20mg of Tirz in 1ml of bac solution. To inject 2mg of Tirz, use 0.1ml (or 10 units on an insulin syringe) of that solution.
To answer your question, yes you can add 1 ml bac. You would then inject 20 units (0.2ml) of your recon. Adding fresh bac (with benzyl alcohol) to your vial will boost it resistance to new bacteria. At 2mg, you have 10 doses, a 10 week supply. That will be a total shelf life of over 4 months. Ouch.
Your goal of a one to one ratio is fine but is not a common medical target.
Manufacturers normally dictate how much bac, nurses need to use.
Lilly use 50units of bac per Tirz injection. Vaccines are often in 50 units of bac. Are more diluted. The reason: you 'may' get a skin rash when using only 10 units of bac for that 2mg Tirz dose (quite concentrated). But many folks use 10 or 20units, and are fine. I always inject doses suspended in 40-50 units of bac to avoid swelling.
Where was the vial stored? Room temperature? My compounder said max 3 weeks shelf life at room temperature. Refrigerated? then 60 days, from a compounder (using a sterile environment) seems fine, IMO. The expiry dates on mine showed one year away.
I use 60 day old peps often. Up to 90 days sometimes, in a not very sterile environment.
You could filter it into a new sterile vial to get 4 months. I have read some folks go up to 6 months. It all about sterility. And courage.