Many years ago, when I was a penniless student and had better things to spend money on than accommodation or tents (such as beer, food, caving gear, and transport), I used to go caving in the Dales most weekends, and bivvied most of the time, equipped with a yellow Karrimat, an orange plastic survival bag, and a -5° rated hollowfill sleeping bag.
Use such shelter as you may be able to find. I used to know where most of the usable derelict barns etc were, and quite often used the pedestrian underpass under the A65 at Clapham at one stage.
Sleep on top of the survival bag unless it's actually raining, or windy/cold enough that the sleeping bag on its own is inadequate. If I did get inside, the sleeping bag soaked up enough condensation that it took about 3 days to dry out (OK as it was only used Saturday nights).
One lasting memory is of a milk tanker driver stopping to express his astonishment when he saw me standing dressed in T-shirt and undies in about 6" of snow on Ingleton Common. He'd come along just as I stood up to get dressed, with the sleeping bag out of sight round my ankles, me having picked enough of a dip that I wasn't visible when lying down, as camping on the common had recently been banned.
If you do use a proper bivvy bag, it's very useful to also have a small tarp & support. If you don't have any additional shelter, and it's raining hard, it's very difficult to get your sleeping bag unpacked & into the bivvy bag, and you undressed and into the sleeping bag, without getting the sleeping bag wet enough to make for a pretty uncomfortable night. Cycling waterproofs are nowhere near sufficient shelter.
If you want to be reasonably comfortable, there isn't going to be much of a weight difference between tent, bivvy & tarp, or hammock & tarp, and probably not much of an overall cost difference either.