Wuh oh, you must have purchased two different versions there on accident. The 45$ version should give you access to the beta and it should download (granted that there's nothing going on with itchio's platform or launcher).
But only if you payed for the 45$. If you have purchased both tiers: the base 29.99 and the 45$. I recommend contacting support via itchio's email service: support@itch.io ( I believe that's the email I used when I upgraded to a different tier. It took them about 2 business days to reply).
They should refund you one of the tiers. If you still want the beta access just make sure you ask them for a refund for the 29.99 tier and tell them you made a purchase for the 45$ beta tier.
Sorry if it's not downloading the 45$ tier you purchased. Something might be wrong with their platform or itchio's launcher app today. Because I tried to update to the most recent beta version and it's still not working. But, I already have the game installed, and have been playing the beta. So for me, I don't know if it's the app or the file, etc. But you should def try and contact support so that you get a refund for that 29$! Hopefully Meyaoi can help you if the issue persists.