Dissertation reader issues
Background:
My similar situation
I did have a problem with my committee. I started a tenure-track job before I defended, and to my horror, one committee member resigned prematurely. I was a two-day drive from my alma mater, and had to find someone very quickly who was both qualified and willing. I proposed a tenured professor in a field closely related to mine, with similar research interests approached by the methods of his field. He had a good CV and was therefore acceptable to my advisor and the rest of the committee. He had a mild crush on me at the time, though not a serious one, mainly because I was the new kid on the block. That might be why he agreed to the thankless task, but I couldn't concern myself with his reasons. He came through, which was all that mattered, although there was a brief crisis after I had made it clear that I wasn't going to participate in any improper flirting with someone on my committee and found my unmarked dissertation shoved under my office door. 

A cautionary tale 
The committee member who quit had been extremely problematic while I was still in the department, to the degree that I went to the ombudsman for graduate students to discuss the matter and get advice. I was assured that the meeting was entirely confidential, but soon learned that it was reported to my department's chair and made its way to the reader. Therefore, I advise caution in going outside the department.

Answer:
The money issue
You have two questions. One concerns the possible financial cost of a delay caused by the recalcitrant reader. That can be taken up with the university's administration, starting at the lowest conceivably helpful level which would be somewhere above the cashiers. In asking about it outside your department, you should relate very little about the reason for the delay. It should be sufficient to say that it has to do with with the schedule of one of your readers. After a face-to-c=face conference with the appropriate authority, ask for leave to submit your case in writing. Lay it out in calm, practical and professsional language. If you're refused a waiver, write again and politely requested references to the written policy that governs. If there is none, you have some leeway to appeal the decision and assert the need for an accommodation in cases like yours. You might end up pretty high on the chain of command, so tread carefully. The higher you go, the more likely it will get back to the problematic reader, so each time you have to go over someone's head to have their decision overturned, you  should weigh the pros and cons of opening a can of worms. (I don't know what the tuition cost is; your decisions about pursuing a waiver will obviously be affected by the affordability of staying in school longer than you planned to.)

"Self-help"
If you want to go to the reader yourself and ask when they'll complete their review, it's essential that you have your advisor's permission to do so. To do otherwise might create the appearance of "going rogue." The risks your advisor mentioned, associated with involving the department chair, probably exist because your advisor sought help from an authority, which is about as popular as going rogue.

Are we there yet?
I interpret "it will take a while" to mean more than two weeks, with anything up to two months possible. 

A practical proposal
If there's any chance to replace the problematic reader with someone as well qualified but not as slow and uncommunicative, that might be ideal. It would be polite, with your advisor's knowledge and permission, to ask the problematic reader if he or she minds being replaced.

Philosophizing
Fact is, a student shouldn't have to handle this alone. Your advisor should be capable of resolving it for you. 

The other fact is, your advisor can't control a tenured professor, if that's what the problematic reader is. Neither can the department chair. Maybe a dean could, but this isn't a matter for a dean.

Giving up all hope and the rewards that can follow
If you end up having to pay for another term, make the best of it by keeping a toe in the research you're interested in. If you can help out in your advisor's lab, do so. You can also analyze and write up any worthwhile unpublished data you collected as a student, or work on a scholarly review of research in your narrowly defined area, addressing questions and problems that remain unresolved, if there hasn't been one recently.