If you have a major concern, you may want to look at taking antibiotics possibly. But you can always contact your OB/GYN and discuss with them about your gum problems. Your OB/GYN may authorize your dentist and clear you for dental treatment. There's no reason to avoid some minor scaling and root planing in the area if you're in pain and its an emergency. If its not symptomatic, then I would agree and wait until the second trimester.
Unfortunately hydrogen peroxide wont treat infection. It is always best to consult with your medical doctor before introducing any new chemicals into your system while pregnant. As you said, protecting your baby is the most important thing. Your medical doctor may be able to suggest ways that you can get some relief without jeopardizing your baby's health.
Best of Luck