X-wing and Armada work under a permissive ruleset, as in you have to be told if you can do an exception to the basic rules.
For example: Darth Vader and Soontir Fel are both at distance 1 from Rhymer, but medium range from an enemy ship (unengaged for the purposes of this example). Major Rhymer is at close range from that same ship but outside distance 1. What attack dice does each squadron roll? Rhymer clearly rolls one black die (his battery). Soontir Fel clearly rolls one blue die (his battery). Does Darth Vader roll one black die (his battery) or no dice at all? While Major Rhymer would allow him to attack at this range, the rules also state that black dice are not used beyond close range. My assumption is that Rhymers ability would allow Vader to roll one black die, but there is a logical argument to be made that although Vader is allowed to attack with all of his dice, black dice are not rolled at medium range. So Vader is able to attack with "all" of his dice, but only rolls those dice that are appropriate for this range. If his battery were one black and one blue (like a B-Wing) it is entirely possible that he would roll only the blue, but also possible that he would roll both.
Once again, I am looking for specific rules that state WHAT is done, and WHY, not an absence of rules to the contrary.
^^^ This is NOT what you initially asked. You asked about whether Rhymer let you break Engagement rules. Nowhere in Rhymers card affects engagement, thus the reactions to your question. A permissive rule set means that Rhymer would have to address engagement specifically in order to have any effect on it at all. I know personally i was thinking WTF Kinetic, this is some bizarro Rules Lawyer BS you're tapping into.
As far as your above question, lets break it down.
Standard rules-
Attack Range: Attack range is the range at which a squadron or a ship’s hull zone can perform an attack as determined by the armament it is using.
• Each squadron’s attack range is distance 1.
Rhymer Rules-
Friendly squadrons at distance 1 can attack enemy ships at close-medium range.."
Rules Result: Rhymer clearly changes all eligible squadrons (those at distance 1) from a distance 1 to close-medium range.
Standard Rules-
Roll Attack Dice: Gather attack dice to form the attack pool and roll those dice. Gather only the dice that are appropriate for the range of the attack as indicated by the icons on the range ruler.
◊ If the defender is a ship, gather the attack dice indicated in the attacking hull zone’s or squadron’s battery armament.
Rhymer Rules-
"... using all dice in their battery armament."
Rules Result-
Eligible squadrons can now gather all attack dice in their battery armament regardless of the (formerly) appropriate range as Rhymer exempts them from the close/medium/long.
In conclusion:
Rhymer allows all eligible ships (those friendly and within distance 1 from himself, including himself) to fire at up to medium range, using all dice in their battery armament, irrespective of the typical range restrictions (Black only within Close Range, Blue only within Medium Range.).