There are trimmers available with engines larger than 31 CC, but they tend to be heavy duty commercial models that cost considerably more than $350.
I've used a variety of brand name products in our groundskeeping division over the last 20 years and have found Stihl equipment to be the most reliable and useful. It also costs a little more.
From the description of your intended usage, I think a light duty commercial machine such as the Stihl FS 90 would more than meet your requirements. It's a straight shaft model that has the option of being outfitted with a selection of steel cutting heads for brushier areas. It should come in at around $340.
You can check it out here:
FS 90 Bike Handle Trimmer - STIHL USA