The implication of your post is that you’re struggling to get to 1500 calories, but you’re also trying to lose, presumably, a large amount of weight.
If you’re overweight, you clearly know how to eat enough calories. Eat more, like you were doing when you became overweight in the first place.
If you’re not overweight and you’re struggling to eat more than 1,000 calories, you should probably see a therapist about a potential eating disorder.
More broadly, eating 1,000 calories can make losing weight harder because you are likely to lower your basal metabolism and giving yourself less energy to burn calories through activity.
The math of 1,000 calories/day works out theoretically and may seem enticing (“I will lose an entire extra pound a week!”), but in practice it can often make things more challenging than it needs to be.
The simple fact is that losing weight is a long-term process. And, in general, you can gain a lot more weight in a month than you can lose, so weight gain/loss are not symmetrical processes.
In terms of your specific question about “eating back” calories from exercise: in general, you should indeed increase your calorie consumption if you are regularly exercising. Whether you should eat back every calorie you burn is far too nuanced a question related to exercise routine, health goals, basal metabolism, diet, etc. to answer in the abstract.