According to the Prima Guide for the game, while you can build more bridges for the town, there isn't a public work that allows for more beach ramps.

Does that mean that I shouldn't give my highest scoring bugs first?

Yes. If you want to get as many items as you can get you should start off weak.

I'm also concerned that if I give lower quality bugs that I'll get lower quality furniture, or will Nat always give you a piece of the Bug set regardless?

How it works is until you meet a certain point level (it was 80 points for me) you will get normal furniture from him every time you break the record. After you break that level of 80? points, each new record after that (including the first) will yield another bug furniture.

So I'm not even entirely sure I can direct the outcome the way I want to but I'd like to try. What should I do to guarantee the most pieces of Bug furniture?

Bugs are graded by three things: rarity (easiest to "manipulate" but you have to find the right bugs), relative length (the game tells you this right when you catch the bug, if you can sort your bugs of the same type by length) and luster/color (this is random and there is no indication of luster until you give it to nat). To maximize the bug items you get, start with not-so-rare bugs and work your way to the rarer ones, working in order of size when you have a bunch of the same bug.

Rarity can roughly be measured by how often you see a bug around. Common butterflies are super common, tiger butterflies are uncommon, emperor butterflies are sort of rare, Raja B butterflies are quite rare, as an example.

Note that the bug off is held monthly, and you probably won't get all bug items in one month. Personally I only managed to get two (I could have gotten more though).

Also not that at the end you'll get a ribbon based on performance. You get a ribbon based on placement, not on how many bugs/furniture you've gotten. A single, high scoring bug will get you a gold ribbon.