They'd still feel like very different games, even if you had 80 - 100 figures on each side - which is pretty much standard for Billhooks and roundabout what you'd get by doubling up the usual LR points total.
I have played (and am playing) both. They both give entertaining games, with that 'emerging narrative' feel that many of us enjoy, but they are quite different sorts of games - although they do also share certain similarities.
The good thing is that both (mainly) use units of 12 and 6, so you can build and use the same forces for both sets of rules - although in Billhooks, cavalry are in units of 8 not 6.
They are both 'bucket of dice' games, and share certain other similarities. Each leader has a command radius of 12" etc.
I suppose the main differences are that:
1. Lion Rampant (and all its offspring) provides a much more freeform game. Troops are basically blobs of figures that can face, move and shoot in any direction, and cope easily with terrain and obstacles. A game of LR benefits from a table stuffed with scenery. Billhooks is much more restrictive. Units (except light cavalry and skirmishers) are strictly ranked up and can only move carefully and slowly in anything other than straight ahead. Wheeling, turning and changing formation takes actions and can cause disorder - which then takes more valuable / scarce actions to recover from. Ditto crossing obstacles. Billhooks is really designed to be played on a largely open field. It's all about manoeuvre. Whereas manoeuvre in LR is hardly a consideration at all. You just move where you want and how you want.
2. LR is more abstract. It's a set of generic rules that can be used with almost any setting or period - witness all the spin-off editions for different genres, which are essentially all the same thing with slight stylistic tweaks. Billhooks is purpose designed to provide a reasonable approximation of late medieval force composition and tactics. It has some nifty little rules like 'arrow supply' for your longbowmen, which means you have to think about where and when you use your resources.
3. Billhooks unit activation is card driven (like Muskets and Tomahawks if you've played that). The last card each turn is never used, so one leader (and his units) are likely to fail to activate every turn (although it can be one of the random bonus cards that is the last one remaining). Because of the randomness of the cards, activation usually switches backwards and forwards to either side within each turn. In LR, you can activate all your units in each turn - until you fail one activation roll, then that's your whole turn over. So if you fail to activate your first unit, your whole army of seven or eight units is stymied. Many people hate this. I quite enjoy it
It tends to even out during the course of a game.
4. Although both are blessedly short and (at first sight) simple sets of rules, in the playing, Billhooks is much more sophisticated. It's a simple set of rules but with many cunning wrinkles and many little details that keep you on your toes. Put it this way - I've now played over a dozen games of Billhooks - which I like very much. But there are still things I get confused about and get wrong in every game. Once you've mastered LR on the other hand, it's essentially what you see is what you get. It's a way simpler and easier game to play. But clearly less 'realistic' (insofar as any wargame is ever really realistic).
5. A lot of the character detail and period colour in Billhooks comes from the leader and event cards, which introduce more random elements. In LR, you can build in some of this sort of stuff by using leader traits and so on. But you can have a perfectly straightforward game of LR without using all those optional rules. In Billhooks, this stuff is more baked in.
Honestly, they both provide very enjoyable wargames, and I like them both for different reasons. Both are scale-uppable in my experience, beyond what either set of rules is really designed for. The mechanics work just as well with multiplayer games - they just take longer!