First, heat-shrink is a no-no on Amsteel. Amsteel melts at only 300F and any heat approaching 200F seriously weakens the rope permanently. Reference: http://www.samsonrope.com/site_files/Tanker_Mooring_with_ASB_in_High_Temperature_Climates.pdf I use genuine 3M electric tape available at Home Depot electrical dept. It does not gum up (as much as others).

Second, I would start the tuck on the Modified Brummel immediately after the Brummel. It looks like you started your bury about 4 or 5 strands down, whereas you should bury at the first opportunity. The Modified Brummel does get its strength from the bury (Chinese finger trap) and the Brummel is just a lock that limits slippage. While I agree it would be beneficial to put in a few lock stitches, it's not necessary unless people pull the tail out, as evidently happened when you passed it around. Once the loop comes out like you show the only way to fix it is to pull it all the way out, run some electric tape lengthwise to contain the tapered strands, use a fid to push it through the bury again, then peel the tape back off starting from the non-tapered part, then milk it back inside the cover.

As for not being able to use an eye with the modified Brummel, you can use stainless quick-links, or you can put the entire loop through the hole then thread the other end through the loop so it's captive.