Opening: "In the time period my book series is set, tensions between the Haxamanian and Aurean Empires need to be relaxed (..) is it possible for both the Haxamanian and Aurean Empires to rule Tifinaghia together, jointly administering the region despite having radically different governments?"
My scenario before the joint rule
I haven't read your part 1, so I'll need to reconstruct a version..
Haxamanians and Aureans are both humans. They shared the planet. Aureans are the descendants of an earlier civilization elite, that colonized several near planets. They left Awal, because it was going to be hit by in a meteor storm that would last for centuries. The Haxamanians stayed, they survived and developed a technology level similar to ours.
When the meteor storms eventually set, after some 1200 years, the Aureans returned. At first, their technology was superiour, many Haxamanians died. However, during the storms, Haxamanians had developed very good shielding. The Aurean attempted to recolonize the planet, but they failed. In the long time the war lasted, Haxamanians developed sufficient technology to prevent the Aureans from conquering their planet. After a 120 year fight, the parties decided to start negotiations. A flood that hit Tifinaghia and the subsequent help that came from the (Haxamanian) Inumideni neigbours was the primary incentive to end the war.
How joint rule ideals landed in federalism
Many Aurean civilians realized they longed to return back home to Awal. There were Aurean colonizers and other civilians who bought land. They built factories, resorts.. and economic activity rose, with Tifinaghia remaining the Aurean center of activity. At first, it was a military base, now there is peace and Tifinaghia has become a tourist attraction: the beaches, the famous star ship museum, and its good wine, of course.
After war and negotiations, modern diplomacy and politics kicked in, and despite the fact Tifinaghia is a colony, the parties don't agree on giving it independence, and instead attempt a joint rule. It worked at first, Haxamanians do not have an authoritarian rule. There are Aureans in parliament, who defend the Aurean interest. In daily practice, the arrangement resulted in a weak federal rule, and a strong local rule, like American states e.g. Alaska have. The Tifinaghians would mind their own business, speak their own language, but with a strong trade union and open borders. Also, there are Haxamanian courts located in Tifinaghia, that maintain Haxamanian federal law. These courts are mainly symbolizing the peace: it turns out to be very difficult to bridge the language barrier in law and governing. The Tifinaghia flag has joint Haxamanian/Aurean colors, but Tifinaghians generally don't leave their region and most Haxamanians still regard Tifinaghia as "abroad" or "actually ours".
Comparison case on Earth: Hong Kong
I think reaching a lasting peace over Tifinaghia would be complicated, because of the past war and the cultural and language differences. To maintain piece, I divide power, I don't actually share or alternate power. Compare the situation in Puerto Rico, or Alaska. So my pessimism about joint rule results from Earthly perspective.. is there e.g. a racist bias, that may further complicate affairs.. I did not read your book (yet)
As for "empires": the most obvious historic analogy to the Awal joint Imperial rule attempt was not mentioned yet: imho that analogy is the status of the city of Hong Kong, as recently negotiated between the Chinese and British empire.
At first being a British colony, Hong Kong was imposed British language and British rule. A century later, Hong Kong has had a 25 year transition going toward Chinese rule. During this transition, at least in the first 10 years of it, Brits and Chinese agreed to have a joint rule arrangement. Now, Hong Kong has become Chinese, primarily.
On the long term, there could be little difference between the Chinese and Haxamanian position. Eventually, the surrounding "main land Haxamania" will attempt to integrate the region again. Like the Chinese, Haxamanians regard Tifinaghia as part of Haxamania. Portuguese Macao in South China also became Chinese again.. Joint rule is a great ideal, but it requires overseeable cultural differences. Like Hong Kong, the region of Tifinaghia is very rich, but the difference is: Tifinaghia has a very strong cultural identity. Eventually, it will seek to gain independence, and a new war could arise.