Hello,
How long has it been since you went sale agreed ?
There's no logic to setting what appears to be a tight deadline for an engineer's report, unless it's been dragging on for a while, or the vendor has something they are trying to hide.
Assuming this hasn't been dragging on, I'd schedule an engineer to do the inspection as soon as you can, and notify the sales agent that the engineer will need access on the relevant date, copying the engineer on the email to the estate agent.
I'd forget going with the engineer - you'll only slow the process down, and run up extra costs. If you don't have the expertise to know what you are are looking at, then you could be told anything, and you won't know the difference. If you do have the expertise, then why not go to the property yourself, seperately, and make a note of your concerns, then share them with the engineer, before their inspection? If you don't trust the person that you are engaging to do the report, from the outset, then you need to ask yourself why engage them, and not someone else!
Is there any chance that there's some confusion between the valuer, and the surveyor, here, btw? I'd expect the estate agent to be pushing for the bank valuation to be done, rather than the engineer's report (assuming you'll be getting a mortgage).
If you are really unhappy with the estate agent's behavior, then arrange to meet them in their offices, talk it through in a reasonable manner, and if you don't get satisfaction, ask to speak to their superior.
Finally, where is the Title to the property, is it with your solicitor for inspection, yet? If so, you've a bit more control on timeliness than the estate agent is making you think. If not, tell your solicitor to chase them up, and also email the estate agent to tell them that the vendor's solicitor is now delaying your progress. Once you're solicitor has the Title, get your solicitor to phone the vendor's solicitor, explain that your are arranging an engineer, and want to progress ASAP, but that you are struggling to get an early appointment due to high demand etc. No harm also having them mention that the estate agent is causing problems, if that is the case, and that the estate agent may put the sale at risk, if they don't cool their jets a bit. That message will get to the vendors, regardless of what any pushy estate agent is trying to tell you, and the vendor wants your money!