There is a lot of good advice in this thread. Just to add to it, firstly, as has been noted, a PhD isn't required for most design jobs, and is unlikely to be more useful than a bachelors or masters. PhDs can be useful for R&D and specialist, highly technical roles, but doesn't necessarily trump an undergrad or masters for the rest.
The full five year course is probably the best route to take to be competitive for a graduate role. A year of industrial experience will be extremely valuable, and many big-name or well known employers also prefer masters level candidates (though this is sometimes mainly a way to reduce the number of applications to review).
But whether the masters is an MEng or a standalone MSc doesn't usually matter for the purposes of employment. I would imagine an MEng in bioengineering would already be highly specialised, though it's not uncommon for those studying the traditional engineering disciplines to do a more specialised MSc after their bachelors. An MSc may also be cheaper than an MEng - though it'd have to be funded out of pocket. It may be useful to weigh up the financial pros and cons of the MEng vs an MSc.