The cap would only be charged to a hazardous level if you tried powering the circuit without a load connected. Otherwise, a bleeder doesn't help anything here. The diode itself clamps the cap voltage to operating voltage or below.
1K is very low for a bleeder anyway. If you've got 10µF (any more than 10µF isn't necessary for lm317), the cap is discharged to <14% original voltage in 20ms. 100k would waste 99% less power and discharge to <14% in 2 seconds, which is plenty fast enough. 100µF with 10k would waste 90% less power and still discharge in 2s.
1k will work, but I personally try to be a little more practical than "pick a number, any number"
7.4V isn't enough. The minimum dropout depends on load current and junction temperature, but is usually around 3.5V for what we use them for. You're saying 4.8V from the diode for a total of 8.3V minimum to be in regulation. It's even worse when you take those anemic wires and internal battery resistance into consideration. It will turn on at 7.4V, but it will not be in regulation. The current will be lower than your setpoint, and the regulator is acting as a resistor here. Switch to 11.1V.