Offer on house accepted but with non-refundable exchange deposit? Advice appreciated

Me and my partner are first time buyers and we have just had an offer accepted for a house priced at £360,000 (£15,000 less than the asking price). When submitting this offer, our estate agent recommended that we offer an exchange deposit of £1,000 to £5,000 alongside our offer of £360,000 as there was a few other interested buyers and this would increase our chances of our offer being accepted. We decided to go with a £1,000 exchange deposit and this was accepted by the sellers over the weekend. Terms were drawn up by the estate agent, which we agreed to, and this included a clarification that the final property price would be £359,000 with this £1,000 non-refundable deposit.

Yesterday, I was contacting a couple of conveyancing solicitors to get a quote for legal fees and one of the individuals I spoke with strongly stated that this recommendation was very unusual from our estate agent and may cost us more money when it comes to legal fees and time to get the legal work sorted (as an experienced conveyancing solicitor will have to draft and agree terms for the Lockout Agreement).

We have gone back to our estate agent to ask if the seller (and their solicitor) would consider £360,000 with no exchange deposit to save us time and money. He has come back and instead asked if we'd be willing to place the £1,000 deposit with them (the estate agents) at no additional cost. He said this would formalize the arrangement with a signed agreement to ensure everything is legal and fully compliant.

I have very little knowledge when it comes all of this and I have no idea if this would be entering a legal grey area. We do not yet have a solicitor but the sellers do so I'm just very worried about doing any of this without a solicitor involved on our side (which hasn't been clarified by the estate agent). The estate agents have a good reputation and he has stated that he has dealt with similar situations when it comes to an offer with an exchange deposit but it is just getting really confusing.

Does anyone have any advice or thoughts?