Is it normal to ask to see the entire manuscript (not just the abstract) before accepting a peer-review invitation?

It is rather common in mathematics to have access to the full paper before deciding to review (it is attached or a link is provided). You could basically use what you wrote here as the basis for your reply. I specifically mean the phrase:

I am potentially interested to review the paper, but I would like to see the content to be sure if the methods used in the paper aren't beyond my knowledge.

I would just ask for the full paper with this reasoning.

Only make sure to decide whether you are willing to review in a timely manner after you received the full paper.

I gather from other answers that the norm in math is to send the entire paper when requesting a peer review. By contrast, in the fields I'm familiar with (generally, biology/molecular biology) it's the opposite; peer review requests send the title and authors, and often but not always the abstract; never (or very rarely) the full paper.

I assume that one reason is to avoid conflict of interest awkwardness. Editors presumably don't want to accidentally send the full description of a project to someone who is directly competing.

Edit to add a possible difference from math: I have the impression that math peer reviews are much more time-consuming than in biology, and that a peer review involves basically working through the entire paper oneself. That's rarely possible in biology (I can't take five years and 5000 mice to repeat a set of transgenic mouse experiments) so my peer reviews might take a few hours to a couple of days, and I don't need to see the paper to judge how much of a commitment I'm making.

I referee in math. I would never accept an assignment without first seing the whole article. And, over many years, I have always been offered to see the manuscript, both when asked by email directly by an editor, and also when receiving an automated invitation.

Are you sure there is no link in the email to see the article?