Growing a page is too broad of a goal.
Pages can have many purposes - Engagement, credibility, website traffic or even sales inside of Facebook,
the strategy will off course change accordingly.
if you really want to engage your audience, or create a reliable long term community - that takes quite a lot of time and wont work with Post Engagement ads or Buying Likes.
if the page should just look alive and AS IF it has a lot of activity, while this activity itself is not part of the core audience, you can sure only work on making
the right impression and do well with fake likes, editing dates on posts and time published posts for your future content.
Take in account that mixing those strategies might be the wrong choice if you plan to TARGET those page likers, since they will be mixed with fake ones
and even the ones that PPE campaigns can gather, are mostly fake, or more gently, not the ones to take action on your offers.
You can limit your targeting to Dates where you didnt buy the likes, to try to figure who the real likes are BUT
From my experience Page Likers is usually NOT one of my strongest audiences (mainly because if i target them i exclude purchase or even pageview, so i get only those who liked but didnt take further action)
it depends in what other stages of engagement you Plan and Can retarget those users.
Considering there is a website behind the page, just stick with "Custom Events" like pageview ( even PVx2 for higher intent) of certain pages, add to carts etc, than i wouldnt mind too much about the page audience.