1. Your solicitor is correct IME, €1k / €2k to the EA is more than sufficient as an initial, receipted, refundable, no-contract 'booking deposit'.
It sounds like the EA has reservations about the OP and does not want to lose the underbidder if OP has change of heart. Strong deposit = strong intent, but the OP should only pay what they are comfortable with and c.5k should suffice. it's refundable anyway.
Your solicitor has a challenge with you as a client. I wouldn't countenance parting with big lumps of cash to an EA. Every piece of money after the initial 1k/2k 'booking deposit' goes to your solicitor in the first place to be disbursed from there. Any and all payments from now on are subject to the contract conditions you agree with your solicitor (full structural survey, finance, searches, title, vacant possession, closing dates, etc, etc.)
2. What 'bidding war'? One of the first things you agree with the EA having parted with the 1k/2k 'booking deposit' is that the house if off the market (no gazumping) at the price you have offered for it. Who else is bidding or what further negotiations on price do you foresee?
The EA cannot guarantee no gazumping as it is not illegal and is at the whim of the vendor not the EA, just like gazundering is at the whim of the OP and seems to be on their possible agenda.