So the answer is indeed a heavier yarn. Emma Jane proved it. Although she did confess that she often plays fast and loose with gauge... with sometimes hilarious results... in this case, she got it right.
LK doesn't have Lamb's Pride in the colour I want, so I've ordered it, and will be forced to focus on one of my other existing projects until it arrives.
That's not a bad thing.
What is on my needles right now?
For Boring TV:
There's the Highland Lace Shawl. Requires much attention.
For Exciting TV and Car Journeys:
Requires little attention, but is large and hard to carry around: a feather-and-fan shawl in some hitherto abandoned cast-off Fleece Artist Country Wool.
Small and requires no attention at all: rainbow socks.