Anglers are generally small. If you feed your LPS
regularly you should stay cycled and your LPS
will grow nicely.
Your coral tank will stay cycled even without feeding as coral/inverts still produce waste. Sure your bacteria population will be smaller but it won't be much different than adding a new fish to an established system. The added denitrification demand of a new fish will cause the bacteria population to increase.
The other concern is that all photosynthetic coral contain algae that need nutrients to survive. Some nutrient is needed but too much is detrimental. .03 PO4
& .25 NO3
is ideal for SPS
tend to do better in slightly higher levels.
More than maintaining a "cycle" I would be feeding lightly but regularly to maintain health of your corals. Weekly or biweekly feedings of a filter feeding food such as Reef Roids or Coral Frenzy and/or frozen foods for LPS
would be ideal.