Do them a favor and stop using that typeface

Probably somewhat based on (or rather inspired by) Mostra Nuova Alt A
http://new.myfonts.com/fonts/marksimonson/mostra-nuova/

Looks similar, but the bowl on the lower-case "e" is shorter in Mostra Nuova and there is no serif on the lower-case "d"

I wouldn't say similar - there is quite a difference in quality.

Any further ideas?

Looks similar, but the bowl on the lower-case "e" is shorter in Mostra Nuova and there is no serif on the lower-case "d"

That’s what I meant by saying ‘somewhat based on (or rather inspired by)’

Maybe a poor attempt of copycatting Mostra.

My 2 cents:
If the client can’t provide you with the original font (and as penn already implied this wouldn’t be that bad…), suggest to use Mostra Nuova (or another professional font in this style). Some wouldn’t notice and those who do would get it as an improvement.