3.x, one video works fine, another is unskinned for some reason?

I have a header video that's playing fine, and is skinned properly:

http://www.premierrange.co.uk/TEST-SITE/?force_header_video=1
(the video should appear in the large header area)

Now we're sticking a second video in a button, with almost identical markup to the header video, but I find it's not themed at all, it's just a basic <video> tag:

http://www.premierrange.co.uk/TEST-SITE/?force_clicknpick_video=1
(the video should appear inside the "Bundle Discounts" button in the sidebar)

I don't know much about how 3.x applies the skin .. 2.x used to use the parseAllWhenReady function to trigger all that stuff, but now I'm stumped.

My header video markup, the one that works, actually looks like this:

<div id="headerVideo">
    <video id="headerVideoJS" class="video-js vjs-default-skin" width="564" height="190" controls preload autoplay poster="promo-videos/banners/index.png">

When rendered, if I inspect the elements, I find that the DOM structure has been re-arranged, presumably by videojs:

<div id="headerVideoJS" class="video-js vjs-default-skin vjs-paused" width="564" height="190" style="width: 564px; height: 190px; ">
    <video id="headerVideoJS_html5_api" class="vjs-tech" preload="" autoplay="">
    </video>

So the video-js class definitions have been moved to a new enclosing <div>.

The sidebar video, the one that doesn't work, does not have this trickery applied to it. So, something's broken with the way videojs recognises my sidebar video as one that needs theming. Here's the markup:

<a class='button' href='some_url_here' title='Click Here to start building your product selection'>
    <video id="clicknpickVideoJS" class="video-js vjs-default-skin" width="179" height="101" controls preload autoplay poster="promo-videos/banners/click_n_pick_intro_smaller.png">
        <source src="http://www.ourserver.com/promo-videos/banners/click_n_pick_intro_smaller.mp4" type='video/mp4' />
        <source src="http://www.ourserver.com/promo-videos/banners/click_n_pick_intro_smaller.webm" type='video/webm' />
        <source src="http://www.ourserver.com/promo-videos/banners/click_n_pick_intro_smaller.ogv" type='video/ogg' />
    </video>
</a>

Is there some problem with a <video> tag inside an <a> tag, perhaps?