It’s becoming more acceptable to use non-institutional email addresses to contact collaborators or prospective employers. For example, I use my Gmail account frequently and know several others who this as well. You can create an account for free and even have it forward to your existing email address (or forward your existing address to Gmail).
However, for the purposes of having an “institution” email for academic services, you gave a valid reason for getting one. These do not typically expire after your candidature but it is worth checking University policy on this. In my experience, so do allow you to use them for personal reasons and to continue to access them after you graduate or take up another job. As a researcher, it is in their benefit to support you to do your research, including participating in conferences, publishing on journals, and establishing a reputation with citation tools etc. This means that your email address on your publications should reflect your affiliation with them and you should be able to be contacted with it to support the publication, such as answering questions about published methods and data.
You can dispute their rejection of including you. I did the same when I was a postgraduate student. I was mistaken for an undergraduate and denied an institutional email address. I had to explain that I am conducting research as part of my project which will benefit the University. The status of my course was being changed at the time and there was a lot of confusion about it. I think now those enrolled in it are issued postgraduate email addresses automatically.
In my case, I did not have to involve the Dean. I just had to clarify the status of my course. They handle a huge number of emails each year so there should be no reason that they cannot add an account for you. It will cost them almost nothing. You may need your supervisor to vouch for you that you will comply with university policy for using an email address. As you are not a staff member, it is more difficult to hold students accountable. They may want to ensure that their servers are not used to distribute spam or such things (it’s a huge inconvenience to staff if their domain is blacklisted). There may be some minor paperwork involved. Just ask what you need to do to.
Make sure that you are polite. University administrative staff may not fully understand the details of your course. They handle a large number of students requests per day and likely mostly deal with undergraduates (so they may not understand that the needs of postgraduates are different).