If you've had SAP calculations done to comply with part L1b, the same people should be able to do the EPC.

You usually don 't have to have SAP assessments to comply with L1B, compliance is usualy shown by simply meeting the elemental U-values stated in L1B.
However the EPC should be produced by a new build SAP assessor, also known as an 'on construction domestic energy assessor', using the same type of SAP software that is used for new build dwellings. You won't have to pass the SAP, it is just the method used to create the EPC.
When you enter 'EPC' into an internet search nearly all of the companies will be 'domestic energy assessors' not 'on construction domestic energy assessors'. This means that they are only certified to produce EPC's for existing properties using RDSAP (reduced data SAP), not new build. This is a much simpler process, hence the number of assessors available and the stupidly low prices they charge.
You should be looking at at least £70 and the assessor will ask you for drawings and construction specifications. Enter SAP into the internet search and you should be able to find someone suitable. With the right information they won't need to visit site.
I've just done EPC's for three barn conversions on a site using SAP for an architect client who sent the drawings in CAD so no site visit and he got a good price.

You would need an On construction EPC (aka Full SAP EPC or New Build EPC), not an as built or RdSAP EPC. If you go to the EPC Register (https://www.epcregister.com) you can search for an assessor in your area. Although for an On Con EPC the assessor doesn't need to visit the property, but produces the report based on the plan and specs that you send to him.

but you can talk your way out of anything...

It is possible to do a SAP assessment as an alternative route to Part L1B compliance. If for example you are not meeting u-values or percentage glazing is too high a SAP assessment could be done to show that your new dwelling/extension is an improvement on the standard case. I got this company to do this for my extension it cost £70 and they were very helpful and it allowed me to have more glazing than the 25% of floor area normally required for Part L1B. http://www.sap-and-code-solutions.com/index.html