You could reach Kochi and then take a flight or boat from Kochi to Agatti. Yes the airport is open. Now is an ideal and safe time to visit lakshwadeep. Regarding stay options kindly check via Tripadvisor and MMT. If I get to know about stay options will message you through the same answer.
There are a number of ways you can reach Lakshadweep from Mumbai. By bus, train, ship or plane. You can choose the one that suits you the best. Online bookings for all the options available. Yes, Agatti airport is working. Yes, it is safe to travel. Places to stay in Lakshadweep:
Kadmat Beach Resort
Agatti Island Beach Resort
Island Holiday Home
Kalpeni Tourist Hut
Coral Beach Resort Safe travels 🙌