Use your namesake. But I'm sure you already knew that.
I'm a few chapters into a FE6 Hard Mode run myself (my first run). So far, I'm using Alance (though Lance is already turning out WAY better than Alan for me), Rutger, Dieck, Lugh, and Clarine. Of the first 5 chapters, the hardest parts for me have been the beginning of chapter 4 (you get swarmed with mounted units that are all fairly difficult to kill) and the end of chapter 5 (forced deployment of all your weak units + a large number of enemies with ranged attack and/or very high attack coming at you from all sides + getting reamed by the RNG repeatedly is a really painful combination). I'm kind of lucky because my Lance got very solid Def gains and WTFBROKEN Spd gains (he has like, 15 Spd at lvl 10), so I have a fantastic defensive unit that also doubles fairly often, which was a huge help in Chapter 4 in particular. If you don't get so lucky, weakening stuff with Marcus will probably be a necessity if you don't want to get overwhelmed. (even with my godly Lance, Marcus' Silver Lance has been a godsend at times) In general, take your time and don't do anything risky, and pay close attention to attack speeds. There are some surprisingly speedy enemies; early chapter 4 has a few nomads that were capable of doubling units I was actually using, like Alan, Roy, and Lugh. Never underestimate the damage a single unit can do; their stats are frustratingly high at times. =/
I've also been taking time at the end of each chapter to build my supports (Alan/Lance/Roy triangle, Rutger/Dieck/Clarine triangle, and Lugh/Chad B), and I'd recommend doing the same because they can be very handy. I'm usually very OCD about not using good weapons, but I'm using things like Silver Lances and Killing Edges a lot more liberally than normal if the situation calls for it. For example, when I had 2 nomads on me in chapter 5, Rutger was the only one who could double attack them, but his Iron Sword wasn't enough to ORKO either one. Nomads are ****s, so I just took the Killing Edge to one of them just to get rid of it ASAP. IDK about you, but I'd normally just do without in normal runs to "save them for later", and I would not recommend that here. I'm treating this as much like FE7's HHM as I can.