UK-US stock transfers

I live in the United States and work for a local office of a UK company (traded on the London Stock Exchange). My employer has an ESPP administered by a company called "Equiniti". I wish to sell stock held in that ESPP.

The first problem is that Equiniti charges outrageous transaction fees: a flat fee plus $0.12 per share. This wouldn’t be a big deal if my employer’s shares were trading in the hundreds of dollars, but they’re actually trading around $3. So the transaction fee would be about 4% of the sale. I want to avoid that, so my option is to transfer the shares to a more reasonable broker before selling them.

I have an existing account with Fidelity, a large company that I hoped would have the expertise to handle whatever weird requirements Equiniti threw at them. I was wrong. I tried to initiate the transfer from Fidelity, but that failed because Fidelity expects a "10-digit account number" from Equiniti, and Equiniti declined to provide me with one (there isn’t an account number anywhere in their web portal and their support people refused to cooperate). The only option that Equiniti will permit is for me to initiate the transfer from their side, and they want the following details:

• Receiving International Brokers Swift Code (BIC)
• Receiving Brokers Account Number
• Receiving International Broker’s Name / Phone Number
• EGSP (as provided by receiving Broker)
• Name of Bank in Foreign Country
• Receiving Bank Country

I spent hours bouncing between phone calls with Fidelity and Equiniti trying to get these details. Fidelity found a Swift code for me but wasn’t able to provide an "EGSP". Based on my own research, this seems to be a thing used by banks in mainland Europe, so I have no idea why it might be involved in a transfer between the US and the UK. Equiniti refused to help other than sending me more copies of the same instructions.

Any suggestions on where I might find the information I need, or on questions that I could ask Fidelity that might prompt them to look in the right places?