Is lashing equipment considered part of a ULD?
Normally I would put this as a comment since I retired in 1999, and I don't know what current practice is. However, this is a bit too long for a comment. Also, this is from memory and it's been a long time now, as it's from my experience with load sheet preparation in the late 1980s and the 1990s. If a more recent useful answer appears, I'll simply delete this.

Further, this answer only directly addresses the question's last sentence:


  Listing the additional equipment as Equipment In Compartment (EIC) does not seem to be a possibility.


The airplane does not, of course, care whether the extra lashing weight is tare weight or content weight. What it cares about is the total weight at the ULD position, and that's what the weight &amp; balance calculations (probably done by computer) will use, whether given as one value or two values (tare &amp; content) that have to be added. 

Just adding the extra weight to a compartment without specifying where in the compartment it is would effectively put it wherever the compartment centroid was. That would be correct only if the ULD happened to be at that position.

In actual practice it could be argued that just adding to a compartment rather than a position in that compartment would be acceptable in some instances. For example, let's say you're very close to max takeoff weight before the addition, but the c.g. is well within limits. The ULD locations would have been decided by the software (or manually by the loadmaster) to put the c.g. nowhere near the limits. Adding a few pounds of extra lashing might put you over max takeoff, but it's going to make little difference to the c.g. 

All of the three cargo carriers I was involved with insofar as weight &amp; balance were concerned simply listed the total weight at a position. No differentiation was made between tare and cargo. Thus the addition of extra lashing was simply added to that positions weight, usually by rerunning the software to get a new solution.

The most common reason I observed for adding extra lashing was that when loading the ULD to a position, they found that one or more of the floor locks to hold the ULD in position were broken. For such cases, the loading manuals specified how many locks could be inoperative and how much extra lashing there had to be. Frankly, that extra weight was often ignored, but it was sometimes accounted for.

I don't have a definitive answer as to whether extra lashing should be added to tare or content. That is irrelevant to the airplane, though I can well imagine that there might be a form somewhere that required such a split. If someone had asked me back when I working which it should be, I would have said that it should be added to the tare for the simple reason that the ULD is there to safely contain the content. If for some reason extra containment weight had to be added to make the ULD safe, that would seem to be tare weight to me.

And I imagine that a customer, who had paid money for a certain amount of weight to be shipped, found out that because of broken parts on the airplane he had to pay more, he would not be happy.

A thorough search of the weight &amp; balance manuals for aircraft might yield additional relevant information. Some such manuals for 747s and 767s can be found at http://terryliittschwager.com/wbmanuals.php