Should I be compensated for using my own equipment/software for work?

I work full time for a major fashion retail company, and I've recently been tasked with producing videos and editing them for my department.

Now to start off, I have to say that before this, I have always brought my own laptop to work, but not because I needed to, but because I wanted to and I have more control of my workflow on it. I also use the computer provided by them (2009 Mac Pro). I'm also an Adobe Creative Cloud member for personal use and for my other freelance work.

I took on this new responsibility because I feel that it would be an excellent opportunity to eventually earn a promotion and pay raise. However, they have not put any effort into giving me the equipment I need so that I can complete my projects without having to use my own equipment.

I have asked that they provide me with Adobe Premiere Pro, After Effects, and any other relevant software so that I can edit on my work computer. I have also been using my own DSLR to shoot the footage, but I would much rather be using company provided equipment to prevent damage risk to mine on company time.

My dilemma is, I'm afraid that if I say that I don't want to continue doing the projects on my own equipment and software without compensation, they might give it to someone else and I'd be out of a promotion.

Sure, I already use Adobe CC and didn't purchase it specifically for the new work here, but I still feel that I should be getting additional compensation since they're basically having me do things for them with hardware and software I purchased with my own money.

This new gig has also pushed me to want to build a desktop computer at home, because video editing on a Macbook Air isn't very efficient, nor is it ideal. I have absolutely no way to edit video on my company-provided computer. I'm willing to do what I can to make the best possible content to showcase my work and move up in the company, but somehow I feel as if I'm being cheated in some way.

Am I in the wrong? I'd like to know how I can approach this situation in the most professional way possible with my employer.