You are right: "I don't know" is not acceptable, because it means you are incapable of planning. Your situation is not uncommon. This is an estimating and risk management challenge, which is the essence of all project management.
First and foremost, understand that your plan is a plan. You will not make your plan head on, with zero variance. So get real comfortable with that concept. You are after precision, not accuracy.
Second, planning for the project work is based on the utilization of human resources--yourself in this case--who are not 100% dedicated to the project. While your operations work has variability to it, you need to draw a line in the sand that says you will be committed to the project work x% of time, leaving y% to your operations work. That is the basis of your estimate for the project work. However, since your operations work has variability, i.e., supply will fluctuate with demand, so too will your project performance. This is an organic risk of this type of set up, it needs to be documented, and it needs to be communicated.
Third, your project work, based on your estimates, has a minimum, maximum, and most likley number of hours required to complete it. This is hours, not duration. Depending on where you want to fall in that hour probabilistic distribution, you can calculate the duration. Example, you are dedicated to the project at 20%. The hours estimate is 200 minimum, 600 maximum, and 450 most likely. Since you may want to be a bit risk averse, let's use 500 hours as your target. The duration formula is: Duration = Hours / Utilization, or Duration = 500 / 20%, or 2,500 hours or 313 days.
Therefore, you tell that manager you will complete this project 313 days from now plus or minus a 10% variance. You build your schedule, your track your performance, you watch your critical path, you capture your variances, you escalate early any unfavorable variances, you mitigate where you can, and deliver when you are finished. If it takes you 325 days, you are well within your stated variances and you can consider what you did a success.
If you really want to get sophisticated, you can add to this problem your probabilistic utilization distribution, too, meaning you will be dedicated to the project from 4% to 20%, most likley 12%. Run a simulation between that and your hours distribution and see what happens to duration from a pure random perspective. Then, you can base your duration plan on that result and on how risk averse you are.
Here is a sample simulation results using the numbers above:
This suggests your project will run anywhere from 150 days to as high as several years, but your MODE, your highest probability is 300 days. So set your target somewhere there.
This is a rather extreme example so the results are a bit comical, but hopefully you can see how it works.