Given I now have a client called René, it has become very clear that typing e acute isn’t easy in the default Windows 7 UK keyboard configuration. Switching to the US-International one allows useful shortcuts such as ‘e autocorrecting to é. The downside is that the £ and € disappear. Many other variants are shown on this useful guide.

My final answer was to use the UK version of the US-International keyboard. An older one looks like it works, too (installer needs running as administrator).