few things to keep in mind - first, forage 24/7 is absolutely best (as was already said) and it sounds like you are doing that which is great! Next, keep in mind that beet pulp is primarily a filler (as in little to no nutritional value) so while it will add bulk, it will not help with weight other than primarily water weight.
For your horse, it sounds like she has the hardest time keeping weight on when she is in heat. That tells me that there is likely something hormonal going on. In my experience the best way to address that is with a mare supplement - some people use valerian root supplements, there are some such as mare magic, and a multitude of options out there. What I always say is try what is available in your area/country that focuses specifically on mare hormones to help her be more balanced. Hormone imbalances can cause a multitude of issues and the ability to hold weight is absolutely one of them.
In addition, for feed for a harder keeper I have found that most horses do better with more fiber (which free choice forage will take care of) as well as higher fat feeds. Higher protein I find is best only for horses in heavier work that need the added energy and muscle bulk - usually not ideal for horses with hormone issues or challenges and not ideal for horses that can tend to be on the hotter side as the added protein can cause an increase in "hotheaded-ness".
Overall the most important thing to remember is that every horse is different and numeric values on a feed bag aren't everything. Try different things and see what works for your horse. Supplements (and some feeds even contain them) such as probiotics can help your horse make the most use of the feed they are eating rather than upping feed that the horse's system isn't abe to digest. My thoughts are to try a higher fat feed, a mare hormone supplement, and probiotics, and see how that works, and then adjust from there. And of course, it's already been said, but free choice forage is absolutely the way to go!
Aside: if the forage available isn't the greatest quality, many feed stores offer things like "hay stretcher" or cubed or chopped forage that you can feed in addition to the hay/forage you have available to make up for any nutrients lacking in a poorer quality batch of forage.
Best of luck to you with your mare and keep us posted on her progress!