If it were off a Matchless/AJS single it'd be an E3AR or E3N, with a tapered drive end for a sprocket.
An E3H is what went on the early A7s, presumably also with a tapered armature for a sprocket like other pre-unit Beesa twins? Lucas part # 200303 or 305. E3AR is 200370 and the N is 200802. The N version is easily available new, as is the E3HM (for magdynamo) version with parallel drive end for a pinion and keyway. I don't know if either could be persuaded to suit your dynamo (with some machining work on the M) if push came to shove.
Nor do I know if the drive end bearing retainer plate is the same on yours as on the later twins with E3Ls.
Nor if the later more powerful and relatively easy to find E3L would fit on your machine or not. If it would, it's what I'd do!
Pix would indeed help!
You can bore yourself half to death reading the Lucas dynamo lists and what went on what if you want. Here's one link to them:
http://www.brightsparkmagnetos.com/library/Parts%20lists/Lucas%20motor%20cycle%20parts%201936-57/A03-34%20Equipment%20specification%20by%20machine.pdf. That shows the early A7 as having the E3H.
Or here for the even more boring Lucas dynamo parts lists themselves:
http://www.brightsparkmagnetos.com/library/Parts%20lists/Lucas%20motor%20cycle%20parts%201936-57/E01-11,15%20Dynamos%20and%20supersession%20chart.pdfBasically, rotation and polarity are up to you, regardless of any arrow on the body. It needs to motor in the same sense as it is driven, yes.
As you say, you can reverse the rotation by swapping over the field coil wires, one of which goes to earth along with one brush, the other of which goes to 'F', if using an original type regulator or a modern one like a DVR2 which regulates on the live side. Alternatively - on most dynamos bar some very early E3s where one brush is permanently to earth because its holder isn't mounted on a mica insulator - by swapping the brush tail connections over. Not both though or you end up where you started. No danger at all in doing this. Motoring it the direction you want will also polarise it for whichever earth you choose.
You say you've got 'low volts'. Hope the armature is actually OK after all the hassle of getting it, and I'd suggest you hide a decent regulator in the old Lucas MCR1 box you should have on there, to give the system the best of chances going forward.