Can I use euclidean distance for Latent Dirichlet Allocation document similarity?
Euclidean distance -by which in this application, I assume you mean the euclidean distance in an $n$-dimensional space defined by the distribution of document contents among $n$ topics considered, is a valid measure to use in comparing the topics represented within two documents. 

What you're doing by applying this method is quantifying a topic frequency difference within this newly defined space, and so interpretation of these quanta will require analysis of the space. For example, what euclidean distance indicates that documents are relatively similar? 

In distiction, the normalized result of something like the hellinger distance provides an easily interperable framework by which to evaluate the results- a score of 0 indicates no overlap in the distribution over the topics in question of the two documents, and a 1, perfect overlap.

For the efficiency concerns, it's not clear to me why you couldn't truncate your topics considered to the crucial topics and then calculate any of the metrics on the distributions over ony those topics, rather than the entire universe of considered topics.