I'm with you on the expenses, etc. I also hear you about the kids, having quite a kid collection of my own. (The REALLY expensive thing is when you succumb to the temptation to start buying knives for the kids!)
Here's what I'd do:
1. Get familiar with eBay. If you go for non-fad knives, you can land pretty good deals there. I like your "one little and one big" idea, as that gives you more versatility than trying to get a single do-it-all knife, and also some automatic backup in case one slips into the lake, snaps, etc. For the little knife, consider maybe a used Boy Scout (or Girl Scout) knife, in plain old high carbon steel.
2. Or maybe you want a fixed-blade small knife? The knives made in Mora, Sweden are excellent for this. My best luck on these has been using "Mora Knife" as search terms on eBay, and at
www.ragweedforge.com --go to that page and look for the Swedish Knives category. You can get a very decent knife for your purposes for about $10 US plus a flat $5 shipping charge, last I checked. Consider the Erikssons or Frosts #1 or #2; also consider the Frosts Clipper and Triflex Craftsman. Each of these is about $10-$12, I think, and all are excellent knives.
3. For a bigger chopper, the Khukuri House one might be good--I've read mixed reviews of KH. You can usually score an inexpensive Himalayan Imports knife by lurking at the H.I. forum at around noon (nowadays) Nevada time. Refresh your page often and look for a post from the proprietress, named "Yangdu", that'll have a title like "Blems and Best Buys for 6/11" (look for the date in the title). More days than not, she'll offer a few knives that are either slightly blemished (say, a little rust, or a minor, not-really-weakening crack in the handle material). Lengths are given in inches overall, handle included, so a 15" British Army Service model will have about 10 inches of blade, for example. For your big chopper, you'll therefore not want to go below about 14 inches at a minimum. Upside is that they'll usually come with a functional "karda", which is a small knife kind of like a miniature Mora knife; it's stuck into a sub-sheath in the back of the scabbard. Best news is that the "blems" on the H.I. forum can run as low as $50, give or take--she had a light, martial-arts-style khukuri up for sale, missing its scabbard, today for $45, I think. These prices INCLUDE shipping in the U.S. The trick is that other regulars know this stuff too, so you have to be ready to e-mail Yangdu with your request to buy--this includes which knife, your name and mailing address, and method of payment--before one of the others lurking around in search of bargains does so. The real "sharks" tend to sit on their computers with an e-mail already mostly composed and ready to send as soon as they can fill in the exact description and price.
4. If $50-65 is still a bit much (which, realistically, it may be--I know, brother, diapers are expensive!), you might consider going with just a hatchet for your chopping tool. For an inexpensive but light hatchet, you might consider hitting eBay for a Cold Steel Frontier Hawk tomahawk or Trail Hawk; I'd stay away from the very-heavy Rifleman's Hawk, but that's personal preference talking. For a step up, consider Fort Turner's tomahawks--these are hand-forged, and can be had for maybe $45 apiece; customer service is out of this world.
5. A lot less character than the hand-forged khukuris, but you might also consider a big Bowie for your chopper blade. In your sub-$50 price range, you can probably find on eBay a Western Cutlery W-49 Bowie if you get creative with your search terms and give yourself two or three weeks to sift through the various offerings. You can usually also find someone who's selling an Ontario "Marine Raider" Bowie for under $50 shipped, and that's a very tough knife (much tougher than their lightweight "Survival Bowie"). Another very underrated big chopper for under $50 shipped is Ka-Bar's "Large Heavy Bowie", model 1277. In spite of the "Large Heavy" in the name, it's not all that heavy a knife, but it's still quite tough, and has a nice, flat grind that will let you actually slice things as well as chop. If your hands are tiny, you might find it a bit unwieldy, but if they're ordinary size, you might really like it.
If what you're after is a sure thing, can-buy-it-right-now pair of knives for not much, and you're not holding out for hand-forged artisanship, you could do a LOT worse than getting a Ka-Bar 1277 from one of the cheaper eBay sellers and a Mora of whatever kind takes your fancy at Ragweed Forge.
Good luck! And have fun with that toddler of yours!