It gets costly to use metal clad for such short runs since they usually wont cut it for you, you have to buy a whole roll. Many just use non metallic cable (NM-B or Romex is a brand) on the board with the clamp style box connectors. You will want #10 wire for the subpanel to the subpanel, it will likely have orange insulation, 10-2 is all you need for that since you have no neutral you don't need the 10-3. You will use the white conductor for your other phase and typically the ends of the wires used in this fashion will be marked with red or black electrical tape to indicate that it's a hot wire and not a neutral. For the runs off the subpanel you have options, since you want to power three ballasts you have the option of a dedicated receptacle for each. That would max out the panel spaces but still leave the room for adding a few more lights. With only 30 amp capacity you can probably just run a single duplex receptacle for each circuit, that way you can plug in two ballasts per breaker. 20 amp breakers, 12-2 wire (yellow) and get the 20 amp receptacles, yeah they cost more but they are better. you can only ever run 5 maybe 6 lights on the 30 amp dryer outlet so two lights per 20 amp breaker on your subpanel...
Intermatic T-104 timer (DPST 240 volt timer motor) Has to be a double pole version to switch both hots.
Small roll of 10-2 (orange) and 12-2 (yellow) nm cable
9 clamp style box connectors (for the timer, subpanel and surface mount boxes), I think like 9 of them, may want a 3/4 incher for the dryer cord to the timer and 1/2" for all the nm runs.
6 space surface mount homeline panel
3 x 20 amp double pole breakers for the homeline panel
3 x 20 amp receptacles (maybe orange or red) get the ones that match the ballast power cords.
3 x surface mount square boxes (could use two and put two duplex's in one box but it gets more crowded)
3 x duplex covers for the square boxes
Some short like 1/2 inch) wood screws to mount everything to your board
Maybe a cable clamp for the dryer cord, you could use a strap for 1/2" EMT conduit for that probably. The two hole style would offer more stability for this application.
Oh and you will probably need a couple wire nuts in the timer for the neutral and ground junctions. The neutral and ground just pass through to the subpanel but you do connect the ground to the timer, it has a little green screw, a #12 bare copper pigtail is good enough for that screw to the junction where you twist up the ground from the dryer cord to the timer ground and the ground that goes from the timer to the sub panel. It's easier to junction that with another wire nut since it has the stranded cable from the dryer cable. The red wire nuts usually suffice, not sure there is even room in the timer housing for a blue one.