Lawn Mower Blenny and relatives (members of the salarias genus) rarely eat any type of algae other than the "film" types. In other words fuzzy, hairy or leafy types are often ignored. I believe that this is mostly due to the type of mouth structure they have. They have a disk shaped mouth that is more of a scrub brush that is ideal for scrubbing films off of surfaces but not good a biting or tearing filaments of algae off surfaces. Occasionally we will have one that will eat frozen foods but it is rare. Unfortunately, because we put so much effort into have an algae free environment, they often starve to death. Perhaps try a small wedge of raw zucchini or cucumber pinched in a lettuce clip. I have had great sucsess with freshwater pleco species with this method of feeding. Many pleco have a similar mouth structure to the lawn mower blenny and the zucchini was soft enough that they were able to rasp if off the skin.
As per tangs, I think what should be offered to them will vary depending on what type of tang a person has. Tangs have a wide variety of mouth structures so what & how they eat will different from one specie to another. The more commonly kept tangs of the zebrasoma genus (yellow, purple, etc.) have a mouth good at nipping and tearing off pieces of algae or other leafy material. Nori, macro algae, Romain lettuce and similar is well suite to their mouth structure and feeding habits and they often take right to them. I think the same could be said about members of the Naso genus too. Tangs generically referred to as "bristle tooth tangs" such as Kole, chevron and others rarely eat leafy algae, they, similar to the lawn mower blenny, have more of the scrub brush type mouth good at rasping algae films off surfaces. This is largely true for the acanthus genus as well. I think my point is that a person can not make a blanket statement about what type of algae to feed to tangs because of how different their feeding habits are from one genus to another.
On the subject of feeding algae grown in your filter to your fish, I'm not so sure this is the best decision. I'm not a fan of using most species of caulerpa as a means of vegitive filtration because of the risks and toxins it releases into the water. That does not mean that it should not be used, I just choose to use something a little safer and stable such as cheato. Deliberately putting caulerpa into a display tank can be a very bad decision. Caulerpa is very invasive, once it's in, it can be very hard to get rid of and can significantly impede coral growth. Another reason I am against feeding algae grown in my filter back to my tank is that it kind of defeats the purpose of having a vegative filter. We use macro algae as a means of nutrient export, as it grows, we throw it away "exporting" the waste it has utilized for it's growth. All the unwanted nutrients banked in it's tissue is essentially removed from our system when this method is practiced. Removing it from a sump and feeding to the fish in the display tank is re introducing the removed waste back into the system. Seems counter productive especially when there are so many other better quality foods available. This is perhaps comparable to putting a used, dirty filter pad in the aquarium so the fish can feed off of it - yuk
I use macro grown in my sumps to entice a fish that is not eating from time to time but otherwise I stick with better quality foods such as the formula line and items bought at the grocery store.
Joel