What is your line voltage? (what is the output voltage and current)
Yes it will work. You are only getting power form one line not all three. I tried this for a power meter and the end client rejected it because some time one line goes down and they wanted to keep the meter/computer running. So I made a circuit that takes power from L1,L2,L3 and will work if only line is working.
Simple answer is to use a off the shelf, single phase supply.
If you must take power from all there line I can help.
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Not my circuit but this is close to what I did. The use the "brick" power supply you mentioned there Rload is.
View attachment 120433
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I need to go to work now. My design included the neutral and could run from only one line if needed. I will try to find it later.