I will suggest you to check out
https://www.cryptocompare.com/wallets/#/overview to see almost each wallet available for every top 20 coins in the market.
Zcash
Stratis
Peercoin
From the coins on your list, Coinomi for android will support the ones above. For BAT, I recommend you using MyEtherWallet.
If these are not possible or too many separate wallets need to be created I suppose Exchange sites are safe enough, right?
Exchanges will never be a safe place to store your coins. If you can, use a wallet for every coin you want to store. If you don't have any good option, you may do this. But keep in mind that you are always risking all your portfolio by storing them in an exchange.