As a 3.0 newbie, I'm still getting used to the recruiting changes from 2.0 but I've already seen a couple of instances where I've been on a projected D2 player as a D3 and late in RS1, a D2 SIM will show up in the considering list. When this happens, is it worth trying to stay in the battle or should I just move on? What does it generally take to knock the SIM out before they are able to push up to moderate or higher?

I believe the thought leadership says that if you can keep the Sim AI at "Very Low" after their first approach (assuming you are at "moderate"), or knock them down to it at any point, they will give up. Someone will probably be able to give some more nuance to that thought, but I guess the two schools of thought would be:

(1) have some investment pre-loaded into a recruit so that a Sim's first approach is also their last - this is obviously a risk if a human coach comes in because that investment will be lost.

(2) invest when a Sim AI shows up - downside here is that you give the Sim their initial cycle + 1 more that you will have to (significantly) out-effort them on.

I think I probably prefer 1, but invest via AP and Promises so that you don't risk losing recruiting $s. Keep in mind that any given recruiting action they take is going to be more impactful than the same action by your team, all else equal. Preferences in your favor can hedge this by some amount!

Yep, agreed with redsoxrootr1. You have to get them to Very Low ASAP, and if you do, you'll win against them no problem. If you don't push them down early... you're gonna have trouble--the Sim will keep coming and given enough cycles, will eventually overtake you at D3.
As far as pushing them down, if you unlock the player before the Sim, a lot of times the scholarship itself will put the Sim at Very Low (and any other Sim that pops up will also be at very low and dead in the water when they pop up in the considering list later on).
If you unlock at the same time as the Sim, your best bet is if the player has a "Wants to Play" preference--promising a start and at least 15 mins will "usually" knock the Sim down to Very Low.
If the Sim doesn't have a "wants to play" preference and/or the promised start/mins doesn't knock the Sim down to Very Low, then you're going to have to spend money. And that's risky. If I'm going to spend against a Sim as a D3, I want the Sim to already be at Low and then drop a CV and probably 2-3 HVs. The key is you can't underspend, otherwise see "trouble" above. And once you spend actual budget, there's still the risk that another D2 (not a sim) school comes along and takes him anyway.
On another note, when there are multiple sims on an upper division player to start, just know you have to get them to Very Low even faster and might have to spend more money to do so. When I see four sims on a player to start, I almost immediately just move on to another player--I've been burned trying to fight off multiple sims at the start of recruiting.