Yes, you should get the error corrected. Do it now, before you need it. Although not all employers will want to see it, there's a chance that some may, and if they do, they will probably need it quickly.
Generally you should not include a transcript or other official documents with your application unless they are specifically requested. Your resume or CV will have the basic information about your degree. As you say, it adds several pages; if an employer wants it (and some will!) they should say so.
This may vary by country, but when applying to faculty jobs in the US, perhaps 10% of employers required a copy of my undergraduate transcript; enough that I couldn't afford to ignore it.
I'm not sure what you mean by a "certificate"; the US may not have a direct equivalent. Here you do get a diploma, but this is just a fancy piece of paper that you frame and hang on your wall; it lists only the degree and the date, with no further details. Nobody has ever asked to see my diploma. For us, the transcript is the only real official document; it is sent directly from the institution so that it can't be tampered with. (However, many employers will be satisfied with a simple photocopy for a preliminary application, and only request an official copy in the final stages.)
As there have been some high-profile cases lately of people claiming degrees that they didn't have, I'd expect that in the future, more employers will want to see transcripts.