A fair number of big influencers may not want to jeopardize the authenticity of their account. For those, there is another interesting solution.
When looking at the notifications section of Instagram there is the option to select, "following" and see all of the activity of all of the accounts you follow.

If a big account doesn't want to sell shoutouts, you may be able to convince them to occasionally like your photos.

Initially I was shocked at how many people will notice this by looking at the actions of the people they follow in their notifications. I've seen accounts pass as many as 100,000 followers in one month. People do use that section, and a lot.

You may be able to find a large account that doesn't generally like any photos, and pay them a much lower premium than you would have to pay them to post on your behalf.