Jens Larsen has some great videos, I think that in this case he didn't have time for a comprehensive answer to your question, and instead opted to justify why he made the choice in his video to play it as D Harmonic Minor rather than D melodic minor, given that in the video he is really trying to emphasise how he's trying to create lines that really emphasise the movement from chord to chord.
It's absolutely true that you could use D melodic minor. This fits the A7 chord nicely and then on the second measure you can shift to D dorian ready for the ii V I. D melodic minor's raised (or natural) 6th degree is a B. Compared to the chord we are 'outlining', A7, this is a 9th. So if you play D melodic minor over the A7 you will be 'brushing past' the 9th on your lines. This is absolutely fine and will sound nice and useable, no worries.
Jens' lesson however is about making your choice on any particular chord 'propel' you to the next chord. In this case, making the A7 an A7b9 gives a slightly more 'high tension' secondary dominant, which wants to move a little more to it's tonic. As the A7 or A7b9 is acting as the V (secondary dominant) to a MINOR chord (briefly, a minor tonic), not a major chord, the addition of the b9 is more in keeping with what we tend to do when playing V i to a minor resolution. The reason is a normal minor ii V i naturally has a b9 on the V chord (it's because it's a phyrigian chord with a raised 3rd to make it a dominant chord).
So Jens' logic is that if we are temporarily treating the Dminor as a tonic, before turning it dorian and carrying on with our ii V I in C, then it stands to reason we would use the V chord that we usually use as a basis for a V i in a minor key, ie a 7b9. I say basis because you can really extend and morph this chord much more as your confidence increases.
You can easily create this A7b9 sound by including a Bb, the A7's b9. By turning the B in your D melodic minor scale to a Bb you now turn it into D HARMONIC minor, and include the tension tone that tends to push a V towards a minor tonic, in short you create A7b9 > Dmin.
For creating a line that pushes more towards the next chord this addition may be welcome, however, the choice is yours, and you can achieve a slightly more floaty and slightly more gentle cadence from the A7 to the Dmin by playing D melodic minor instead. This means your A7 chord contains the 9th and this tends to sound smoother than the b9. The choice is up to you, though Jens is on the money in terms of what people tended to do in a 'classic jazz sound' sort of context!
ADDITION; I can see on re-reading my answer how mine and Jens' proposed approach differs from yours. You were trying to find a way for the A7 to fit as much as possible with a following Dorian chord, and our approach was trying to make maximum use of the tension available by temporarily treating the D as if it was a minor tonic. This sort of thing happens a lot, with the 'arrival chord', Dminor in this case, shifting to something else at the moment we arrive. In this case, We set up an A7b9 to propel heavily to a tonic Dminor chord, but when we get there the C# relaxes to C, the D dorians b7, the Bb 'resolves' (sort of) to the B, the D dorians 6. We've achieved more tension and more bounce from one chord to the next, and the voice leading to get to the ACTUAL arrival chord, a D dorian not D minor, is neat and satisfying. Chord to chord movements in jazz are often their own little microcosms of tension and release to better propel through the music, though you absolutely don't have to do this, it just gives a very satisfying, dynamic sound rather than a more smooth and static one.
