(NOTE: When responding, please don't include spoilers for further story content. This is as far as I have gotten in the game and may be as far as others reading a thread like this have gotten.)

Templar, Level 33, Attributes = 19 M, 13 H, 0 S.

I was utterly clueless the first time I fought her, but figured out the "OMG, healing orbs! Kill them, KILL them!" thing on my second try.

But I continued to have trouble on further attempts, even knowing the "trick". As a Templar, I don't have much in the way of wide area of effect, which made it hard to take down all the healing orbs. Plus, being a Templar means being constantly out of magicka in long fights because we have no resource management. With no resource management and no low magicka offensive spells, it is very easy to waste a lot of magicka killing off the orbs and be left with nothing. Then you are in big trouble.

What eventually worked for me as a Templar:

1). Use as little magicka and stamina as possible actually fighting the Lyris Doppelganger. This sounds crazy, but bear with me.
2). Use magicka to heal yourself and only to heal yourself. Templars are good at this! Take advantage of it. We don't have to worry about running out of potions.
3). Don't wait until the last second to heal! You will be stunned/immobilized frequently and unable to cast spells. Hoarding magicka to heal will not be much use if you are stunned and unable to cast the healing spell as you approach death! Keep your health up so you can survive through the stuns.
4). Use stamina to dodge Lyris Doppelganger's attacks and hunt down and kill the healing orbs. I used vanilla bash (right mouse + left mouse, not the Power Bash ability). The orbs go down pretty easy, but there are a lot of them. If you are not using 1H+S, you may have enough power to kill them with a light attack and use no stamina! Good. That leaves you more stamina to run and dodge.
5). Radial Sweep does not use much ultimate, only 75. I only have to kill 2 or 3 orbs to recharge it. As soon as it is charged, I turned toward Lyris (because I have the morph that does more damage to enemies in front of you) and hit R. It'll likely take out any orbs you missed, too. Don't use Nova. It does a lot of damage, but costs too much Ultimate. Radial Sweep actually does more damage per ultimate.

Repeat until she is dead.

Basically: Magicka to heal, Stamina to kill the orbs and dodge/run, orb killing to recharge ultimate, Radial Sweep Ultimate to kill Lyris Doppelganger.

Good luck. It takes forever, but I could not find any other way to kill her as a Templar.

Oh yeah, after I beat her, I followed the arrow and swear it took me back the way I came! When I got back to the Harborage and realized I had come out the way I went in, I panicked that I was going to have to kill her again when I went back. Thankfully, she was still dead when I returned. I have a lot of patience, but having to kill Lyris Doppelganger *again* might have broken me. :-)