Sent Eth from Binance to Bittrex but used the wrong network! :( How to recover?

Transferred my eth (300usd worth!) from Binance to Bittrex but I got confused with all the network stuff and accidentally picked a network (Arb) that Bittrex does not support (they only support ERC20) . :(

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Both Binance and Bittrex said they can't help and I feel like im left in the cold. Have I lost my money? I'm pretty new to crypto and I can't believe how easy it is to lose so much money using crypto with one mistake like this.

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Hoping there is a way to recover this :(

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https://preview.redd.it/5nqqsujge9p81.png?486&format=png&auto=webp&s=8014c593394b9e2e3312c1d1d26f0ceb00b364f2

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UPDATE after all:

1. Ultimately, bittrex is in control since the money is in their wallet but they wont do anything about it
2. I'm disappointed how stupid this thing is, I mean I now understand that there are network options that are important and must be considered (not like an optional thing), but the thing is, imagine sending ETH from one wallet to the other, both wallets correct, and you can still lose your money just because of the damn network option. I can imagine how many people may have lost money this way