This isn't necessarily true, the rule could be that you can't create a double spend.
If TX1 is in the fast-chain, then transactions that spend any of the inputs into TX1 cannot be added to main chain blocks.
There would be no problem with leaving TX1 out of your block as long as you don't include a double spend of any of TX1's input.
The fast-chain would have to be fast, signature checks could be skipped. The owner of the UTXO would be the only one who knew the public key. The only thing that would need to be checked would be that the public key provided hashes to the address.
The fast chain could restrict itself to only standard transactions. Using more complex transactions would be slower.
There is a risk that when a transaction is broadcast someone changes the transaction, since they know the public key (and signature checks are skipped). Some kind of fast signature would be useful.
A new "fast transaction" standard transaction could be added. This would include a 4 byte nonce.
The rule could be that fast transactions must have a number of leading zeros in their hash. If someone modified the transaction, then they would have to re-do the nonce updates.
The number of zeros could be dynamically controlled to try to keep spam low. It should be low enough that it takes < 10 seconds to solve on a mobile device.
Is there an actual signature algorithm which gets high speed at the expense of lower security? I guess any algorithm would work if the key size was lowered?
Could an ECDSA 64 bit key give 5-10 mins of security and be very fast?