"I feel like all of the time and effort I've put into the class has been totally discounted for a reason that has nothing to do with my knowledge of the course material."
Yes, it has been discounted. And, yes, the reason for that discount has nothing to do with your knowledge of the course material.
"Am I wrong to feel as though this is an unfair mark?"
Whether it is fair, or not, may be a quite highly subjective question where people may be very opinionated. Regarding whether you have a right to feel certain emotions: you may.
dropping a grade [such a large amount]
small reason feels absolutely unacceptable to me"
Again, you're welcome to feel however you like. However, such a requirement was probably not created for the purpose of being popular with students. So, frankly, you aren't expected to feel like this is acceptable. However, this is likely beyond your power.
"How should I approach this situation to help repair my grade?"
First, don't just try to figure out how to proceed, based only on your desire to reach a certain end result. Before going through the effort and dealing with the stress of even trying, make sure you're understanding your situation and what your likely obstacles are. Before looking further at "how should" you proceed (which is what you actually did ask), first consider looking at: should you proceed?
Realize that you may not be able to get the desired adjustment, and that may be due to circumstances beyond your control. Circumstances may even be beyond your instructor's control. This academdia.stackexchange.com question (that is being looked at right now) seems to have spawned another question, “Why do some instructors care so much about attendance?” Some information on that question may describe some of the position that your instructor may be dealing with. (Specifically, the information I added to my own answer discusses why the instructor might not even have some flexibility in this matter.)
Then, if you decide you wish to proceed, affecting your grade will need to be done by someone with the proper power to do so. Most likely your instructor will be involved. If you wish to have the college take action based on your instructor being unreasonable, then you might need to involve some other part of the college's administration. That may result in some other staff getting the instructor's perspective. So, don't expect that this is going to happen without your instructor finding out.
Start by determining why you missed class, and whether any of those reasons seem rather justifiable. (In other words, were were they caused by some sort of officially-backable causes, like a medical reason tied to a doctor's note?)
Check official writings, especially including your class's syllabus but also your college catalog, for details about attendance policies and final grades and how you may be able to challenge undesired results. There might already be policy. If so, that policy probably will need to be followed. The policy might be very favorable or very unfavorable. The only way to know is to look up what policies exist.
Once you understand the process as best as you easily can (at the start of the process), come up with the best arguments that you can, so you have the best knowledge you can imagine. You may be tempted to feel empowered by seemingly-compelling logic of your own favorable arguments. That may temporarily be okay, in your effort to get yourself as prepared as possible.
After all that is done, prepare to approach the situation humbly. Start by contacting the instructor to see if mercy is available, or some other recourse (like if the instructor has a pre-designed way to achieve some lost points). Going through the instructor is most likely to be the easiest route.
Alternatives may be more troublesome, which could unnecessarily add workload (to yourself, to the instructor, and to other staff) and may even be a bit offensive, particularly if you didn't even bother to go through the expected common process of trying to work it out with the instructor first. If the results of contacting the instructor did not go as you had hoped, consider whether you wish to just accept this as a costly learning experience (many learning experiences do have costs), or if you wish to pursue some other alternative. (This is where you need to check into the policies of your "student catalog", if such policies exist.) If you can't find any such policies, consider contacting a department head, and/or a college dean. A friendly receptionist might also have some advice about what other resources may be available.