Don't think you'll get withing budget and will struggle with availability but have a look at Puri Mas & Qunci Villas. Just hit your favourite online agency and see what they can offer availability wise.
But whatever you get in Lombok will be 30% better than what you'd get in Bali for the same price.
South Lombok is changing - might be possible - I havn't been but the Novotel Lombok in Kuta (nothing like the Bali Kuta) may be close to budget.