You should recognize why sites try to prevent exactly what you are doing. While you are a member, what if you were not? The majority of visitors to most sites are probably not people, but bots. Lots of sites go to great lengths to make it VERY hard or even impossible to do any web scraping, because they don't know who is on the other end. That's almost certainly what you are running into. The API is supplied as a means to get access without trying to automate the web page, and the fact that they have one means that this is something they are clearly thinking about.

I didn't see any explicit TOS on the site, though I also didn't look very far. Lots of sites explicitly state that you are not to use automated means to login, in which case this site doesn't support doing so, either. Even if the site doesn't explicitly state that, a bit of reading of the link that Plausibly Damp supplied shows that you should expect that.

The ultimate problem with web scraping is that sites can change their layout rapidly, so any solution will break at a moments notice. In this case, the site has posted that they made changes in January and will be making another significant change in February, with more likely to follow. Whatever solution you can get working is probably only going to last for a few weeks, then fail again. That's the nature of the web. Automating through a page is fairly hopeless with a target that is moving as quickly as that site. APIs are provided (usually) to give you a stable interaction. For that site, even their API is a moving target.

Normally, I would suggest getting in touch with the site, as they likely want you to interact with them and would be interested in helping out. With that site....I'm not so sure.