No. Don't try gels.
Your camera should have a custom white balance setting. Something like this:
You need a 3 step process for this:
1) Take a photo of a white object using either Sun White Balance or Flash White balance (I prefer using sun). You can use a sheet of paper but as they can vary in the color you probably need a professional-grade gray card https://www.google.com.mx/search?q=gray+card to achieve a more standardized result. Cheap gray cards will give you problems, and only should be used to calculate the exposition, not the white balance.
Not cheap, but a good one is the one from Xrite. There is a color chart called Propasport that has an additional workflow to standardize your light and colors, called a profile.
You need to shoot that target in close range so it covers let's say 90% of the frame. Don't overexpose it too much. (If you have an incident light exposimeter use it for gray cards)
2) Choose that photo as the custom white balance on your camera settings.
3) Switch to Custom White balance on your camera instead of using sun or flash.
The target must cover almost all framing and lit by uniform light.
Try measuring your source light
Step 1 is the most critical. You can measure these targets I mentioned but also in some cases in the studio you can measure your source light.
If you are using, for example, a softbox you can shoot directly to it, but close the aperture diaphragm a lot so you don't burn out the photo. You need to see that photo in the middle colors. Let's say start with the minimum aperture. An f22 or f32 for example.
If the camera detects that it has zones out of range, it probably will show an error message that you can't use that image.
I'm posting a crude example here. (I choose an old softbox that is not white, but it doesn't fit my flash, I'll update it later)
White is not always white
I'm adding some thoughts here.
You mentioned a white vinyl background. It is important that you use a true neutral gray or true white reference as a reflective target or measure the light source as I mentioned.
Don't use your background to calibrate the white balance. Some synthetic materials have a pink or magenta color, you can think it is white but it is not. If after setting the white balance, you still see the background as pink, it is pink. So probably you need to change your background. If you use your pink background as a white reference, the product on it will have a blue tint.
Don't use a plastic white material as a reference — it is better to use paper or matte painted one.
These settings are saved as a recipe for a raw format and applied directly to the jpg. As I mentioned, you can have additional color adjustments with a color chart and a color profile for a specific lighting situation — in this case, your studio+camera.
To have even more accurate control of the color you need a workflow that includes a color target like a Macbeth chart: Do I always get the same colours when I set the white balance correctly?