During the high season, you'll no longer enjoy be able to take advantage of low season room rates. That said, it shouldn't be difficult to find a place within your budget though you may widen to expand your search to take in venues that aren't necessarily so close to the river.
Luang Prabang is not a big city, and if you stay in the old quarter, it is easy enough to walk to both the Makong and Nam Khan Rivers.
The below places are under the budget and you can also check further check with online agents or the hotel's own web site where applicable, to find exact rates.
The first recommendation is Thongsay guesthouse which has bungalows facing the Nam Khan River and Phousi mountain as the backdrop.
If you prefer to be right in the city, Saynamkhan Guesthouse is a 1939 heritage building on the street corner that offers views of the Nam Khan River from some rooms as well as from the terrace.
Oui's Guesthouse, in the quiet area of Ban Kily, also overlooks the Nam Khan River.
Luang Prabang River Lodge is another good option and is in a prime location overlooking the Mekong.
Another option along the Mekong stretch is Muong Lao Riverside Guesthouse. It is also near the town centre, behind I-elephant restaurant. These lodgings all resemble traditional Lao houses and are within walking of most attractions.