Children have very small airways when compared to us and also their lungs are still developing so you should exercise extra caution and take necessary precautions. You say you have carried the nebulizer and other medication along despite which your child is still unable to sleep. I suggest you visit a local allergy specialist first and find out what actually triggered the asthma attack. Once you know that you will be in a better situation to deal with the problem. From what you have written it looks like a case of dust mite triggered asthma because usually in such patients air purifiers can help a great extent and that explains why your child was sleeping better with an air purifier.
I took a look at the one your are using. I presume the air purifier which you are using isn't very noisy because most of my patients complain that their air purifiers are very noisy.Since your child is already accustomed to this particular one its better you order the same air purifier even if it means getting it delivered to Singapore from the previous place of residence.For temporary use you can use small humidifiers and the allergy specialist might prescribe albuterol for emergency.