I've found that my rabbit eats more hay while in the box. I only put hay in the litter boxes, as she doesn't have a standard cage to attach a hay rack to. I put a handful in each litter box before I go to bed and when I get up in the morning, the boxes are usually empty of hay. At least one is, if not both. So she just sits in the box and eats hay all night.
She also pees in the corners of the boxes so the pee doesn't touch the hay. And the way their poop is, its like "clean poop", so I don't worry about it being on the hay.
So I suggest putting the hay in the litter box. I know you're trying to keep costs low, but keeping your frustration low is more important sometimes!
Here are some suggestions for keeping costs down. You could buy a bale of hay from a farm store, feed store, horse barn. Any kind of grass hay is fine, timothy is harder to find by the bale depending on where you are. Broome hay, coastal hay, orchard grass. Just not alfalfa because she's an adult. You can get a bale of hay for like $4-15 depending on where you are. Right now it would be the second, possibly third cut (if you're in the south), which are fine cuts.
The bag hays from Kaytee and Oxbow, tend to be WAY over priced. Even though I buy bagged hay, I grumble about it every time I do it.
Buying the wood pellet beddings. I'm not sure if you have the wood pellets or the shavings. But the pellets are cheaper, you get like a 40 lb bag for $6 at Tractor Supply and that lasts me, with 2 litter boxes, months. Its well worth it.
Putting hay in the box would definitely help! Just a little handful and that might just prompt your bun to get into the box and go!
Just had another thought reading back through. Is the rabbit peeing NEXT to the litter box? Like she's putting her butt over the edge? My rabbit does that if the litter box is dirty. Or if your rabbit is putting her butt over the edge, the box may be too small. Just a though! :upsidedown: