Sorry, I should have said: sending >0.04 BTC to the bc1 bitcoin address that Breez creates in order to set up the lightning wallet.
I guess you shouldn't try to find out 
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I assume that it's not possible for a (layer 1) bitcoin transaction to fail / be ignored by nodes and minors because of the btc amount relating to a specific address.
Correct.Breez is non-custodial LN wallet
I haven't used Breeze, but I've used Phoenix Wallet. In Phoenix, opening the channel is custodial, but after that the channel itself is non-custodial. If you're just funding Breez by sending Bitcoin to an on-chain address, I assume the same applies.