I'd say don't post your actual ad because we will find you and give you more help than you wanted ;) Like maybe your team page is all white men, or your defence contracts are very prominent, or your CEO is a public dong.
It's a bit of a yellow flag that you don't feel you can go to your managers with this to help fix a team you're obviously invested in? But here's two tips that you might be able to implement or influence by yourself:
1) Treat incoming candidates like GOLD. Respond to them within hours, same day where at all possible. Make decisions really fast, schedule interviews within 1-2 days, drop other things to make that happen. Have future managers & engineers communicate directly, lightly backed up by HR if you have that function (and you trust them). Keep. It. Moving. Make it clear their time is more valuable than yours.
2) Document the whole interview process up-front, put it prominently on your ads, specify the damned salary, and stick to your word the whole way.
If you execute on both of these points, the FAANGs will not be able to come close in a really critical part of the process. They will be unclear about what happens next, who you're meeting, how important it is, maybe it will be a week, maybe it will be 12 weeks, who knows. But they got the prestige and the big bucks, and candidates will put up with a lot for that.
(fwiw, I co-designed careers.bytemark.co.uk/full-process for most of the above reasons, and my current role went from introductory chat->3 interviews->contract signed in about 10 days flat - I like fast recruitment!).