These two posts helped me out quite a lot with a similar concern.
I'm going to give you the details behind the numbers, but keep in mind that you don't have to remember all of this. I just want you to know where I'm getting my numbers from, so you'll see I do know what I'm talking about.
The medical standard for determining your BMR is the Harris-Benedict equation. For women that equation is: 655 + (9.6 X weight in kilos) + (1.8 X height in cm) - (4.7 X age in years).
For you, that would put your BMR at 1960. Then you use an activity multiplier to determine your current maintenance calories. For you, at lightly active, that multiplier is 1.375, which puts your maintenance at 2694.
If you reduce that by 30% (a safe and healthy rate to lose weight), you get a calorie level of 1886 calories per day. Add a little more exercise to that (even some walking each day) and you'll be on track lose a safe and healthy rate of 1% of your body weight per week.
It's not that far off from what you came up with ... but 400 calories can make a huge difference both in the level of nutrients you get and in the amount of energy you'll have to keep your metabolism up.
BMR and maintenance are not the same thing - people get them confused.
BMR is what your body needs to survive if you did nothing but lie in bed all day and breathe. Your maintenance calories are your BMR calories multiplied by an activity factor to allow for your daily activities - everything from moving around, showering, walking around, any exercise you get, etc.
At 180 lbs, it's entirely possible that your maintenance calories are around 2800-3000. A good estimate is to multiply your current weight by 15 or 16 (if you're a man) or 13 or 14 (if you're a woman). That should give you your MAINTENANCE (not BMR) calories.
Also, one of the first things most of us found out when we got here was that we were underestimating our calories, or getting the right amount of calories from not so great foods.
You weight more than me (I'm at about 197) and my intake should be in the 1400-1500 range. That will still let me lost about 2 pounds a week, even if I do nothing but sit still and breathe, which is a fairly safe loss.
Not only that, but my BMR when I was judging it at 215 pounds was around 1600-1700...again remember that this number is the one to start with if you do nothing at all except exist.
Any activity you add beyond breathing is going to burn calories, brushing your hair and teeth, walking to the fridge, letting the cat out, etc.
So the short answer to your question is yes, if you truly are only getting around 1300 a day then you should add a few more so that you don't lose the lean muscle you're trying to build.
But the first thing I would do is compare a few calorie counters to make sure your count is correct. When in doubt, I go with the higher estimates on calories, and lower on calories burned. (For example, I found a counter that says I burn 400 calories brushing my hair, but I have short hair that I don't do anything to and takes about a second or two to brush, so I don't think 400 is accurate in that respect)
To be honest though, trying to remember all of these figures and numbers is a little intimidating to me. Instead I track what I eat and stay within good portions...like actually sticking with the serving size on the package and weighing or measuring out my portions instead of guessing. And I try to incorporate better foods, whole grains, lean proteins, veggies, stuff like that.
If I got this wrong someone smarter will come in and correct it, so don't follow what I said until you see another poster come in and correct me.
But I think I got it pretty close this time.