Stupidly sent BCH to BTC. Is there any hope?

I made the very, very (very!) stupid mistake of sending BCH to my BTC address. To make matters worse, my BTC wallet is on Coinbase, so I guess I don't have any way of obtaining the private key.

My BTC wallet address starts with a 3, so it probably is a SegWit address from what I read. Although I have to admit I don't fully grasp the concept and its implications.

I contacted Coinbase immediately after having noticed my mistake, but haven't heard back from them, yet. The transaction has around 50 confirmations as of now.

I am aware this is completely on me, but of course I would still like to know if there is any chance of me getting a hold of the funds. I saw this post "How to get your recovered SegWit funds using Electron Cash" but if I understand it correctly, this would be something only Coinbase could do for me due to only them having the private key.

I also read that if I found a miner, he could somehow recover the funds, but I'm not sure how and if that works or even how to get in contact with someone who would be willing to do so.

Thanks for listening!

Update:

I just got the following reply from Coinbase:

Thank you for contacting Coinbase about your transaction. According to our records, it appears that you sent BCH to a BTC address. This is an address that can only receive BTC.

In this situation, these BCH funds cannot be recovered. Coinbase does not support transactions where one cryptocurrency is sent to the address of a different cryptocurrency.

For any future transactions, please review your Coinbase deposit addresses and ensure you’re only sending the applicable cryptocurrency to that cryptocurrency’s specific address. You can find your specific Coinbase deposit addresses for each wallet here: https://www.coinbase.com/addresses.

If you have any further questions, please let us know.

Anything I could do to make them help me if that's even possible? :(

Update 2:

After a bit of back and forth, I got this message from Coinbase:

Thanks for contacting Coinbase Support. I understand the importance of your digital currency and thank you for your patience with us.

I've taken a look at your case and am working with a specialist to address your issue properly. Please note that these reviews are typically completed within 5-7 business days.

I won't get my hopes up too much just yet, but at least they are looking into it which is nice.

Update 3:

No real update, but since I got more responses to my post than I could've hoped for, I'd first of all like to thank every single one of you. It's a very unfortunate situation I got myself into, but reading your thoughtful responses and recommendations really lit my day up.

Also, maybe /u/justin_coinbase or /u/coinbasesupport would be able to comment on the matter (case #02990284) or maybe even push this into the right direction?