Hi,
I think transformer based supplies are: VDC = VAC * 1.414, and IDC = IAC * ~0.6. You get more DV volts than you want and less DC current than you would like with transformer based supplies
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LM317 dropout for ~200mA, at 25ºC to 50ºC is around 1.75V, check the graph in the datasheet called Input-Output Differential, or in OnSemi's: "Dropout Voltage", it's not 2.5 times the output power. Presumably you can get away with a 3.7V Li-Ion battery until it drops below somewhere about under 3.5V, and a 9V battery (not the rectangular ones, they last 5 minutes) or a 5V/0.3A phone charger/wall-wart would also work.
Unneeded??? Slight misunderstanding, I think, obsolete means no longer in production, I didn't mean unnecessary, the MOSFET is necessary in your circuit, but the model in the schematic is an obsolete part, okay for one-off hobby circuits but not appropriate for a mass-produced circuit.
Yes, the LM317 is DC current, it comes after the bridge rectifier. Li-Ion have specific charging requirements, otherwise they can overheat and set on fire, read about how to charge them correctly.