My initial thoughts going into the rotation are:
4 Lillie/4 Elm/4 Oak from Remix Bout, depending on deck searching options
4 Cynthia
2 Erika's Hospitality
4 Pokegear 3.0 is mandatory for all decks, I think
3-4 Cherish Ball in GX favored decks. 3 if you have other options, 4 if you don't.
Erika is justified by the fact that decks that still have search options (EM Radar, Net Ball, Mysterious Treasure, Cherish Ball) are still going to have good bench sizes, and everyone else who can't do that are going to run off of Professor Elm or Pokemon Fan Club to fill their benches. The handsize requirement shouldn't be an issue, but it's definitely a card that won't save if you brick so that's why I wouldn't go more than 2.
However, doing some more theorycrafting, I think Lillie is going to fall off a lot as a T1 supporter going forward.
Lillie I think loses a noticeable slice of value after rotation due to Ultra Ball -> Lele -> Lillie no longer being a play option. Even if we did still have Lele and not Ultra Ball, the main thing that weakens Lillie after rotation is no discard 2 from Ultra Ball available to every deck in the game to thin your hand. T1 Lillie plays are going to be on average drawing 2 cards less in any deck that doesn't run 4 copies of EM Radar (so Pikachu & Zekrom GX + Zeraora GX decks), in which case you might as well just play Cynthia really, and it's really hard to justify using her outside of Turn 1 without selective hand discard outlets.
Great Catcher might help her out. Most people will likely be trying to get a Dedenne GX out on their bench ASAP, so Great Catcher -> stick a Dedenne GX up front -> Lillie is still a thing, but that depends on your opponent having a GX target you are willing to pull up, whether you're ready to KO it or not, just to draw 2 more cards from Lillie. This also can't be a play if you're T1 going first since there's nothing to grab.
There aren't really any other draw supporters that I feel are influential enough for me to care about running (Sightseer, TV Reporter, even Tate & Liza to be honest). I think Erika is almost always better than all 3 of those. Cynthia and Dedenne GX will hold you out throughout the early game/set up phase, then Erika is powerful midgame draw power that none of those 3 can match. Even though Erika has a dependancy on opponent's board and your hand size, I would still rather play her since you get to keep the rest of the resources you have in hand on top of drawing up to 5 more cards.