How should our glorious civilization incorporate two neighboring fallen empires

Behold the pure, the glorious, the powerful civilization of the Till Lynesi Star Hunters. We are, without question, the foremost active power in the galaxy in the 23rd century and we will not allow any filthy xenos to contaminate the cosmos, as the divine commands us to ensure. Our neighbors rue the day they achieved sentience as we annihilate their fleets and secure their worlds so we can cleanse them and ensure no additional infestations occur.

While most other empires chafe but eventually cower at our spiritual mandate to fanatically purify the galaxy, there are two ancient empires bordering our glorious realm that appear… indifferent to our strength and tolerant of our presence.

The northern Fallen Empire is a Keeper of Knowledge and is Overwhelming compared to our 39k fleet of destroyers and cruisers. The southern Fallen Empire is a Militant Isolationist and is "only" Superior compared to our fleet. Both empires are equivalent to our naval capacity.

Except for energy, our economy is pretty robust, although we can float on an energy deficit when we undock the imperial fleet because we have tons of influence and not a whole lot to spend it on. In particular, our sectors generate a lot of energy that they don't use which we can cash in on even when they already send 75% of their income to the core systems, and we have a few decent planets for capacity overload. Also, the Energy Nexus technology (Planetary Powergrid) is only 8 months away which should certainly help out, plus one or two other energy-related technologies we could begin researching. Since we can't tolerate any filthy xenos, trading for energy or entering into any other form of negotiation is not possible.

Our military technology is robust as we progress towards our desired ship designs. Starting with missiles (currently T4), we've begun switching over to cruisers with Neutron Torpedoes and T3 Plasma Cannons. We have initial point defense systems and improved bombers to flesh out our offensive designs. Defenses include T4 shields and power cores with T3 armor and shield capacitors. We're currently focusing on building these cruisers, when we decide to build any ships at all.

While we can design battleships, the designs are still early, lacking X-size weapons and long-range artillery. We can fill out a battleship with Missiles, Plasma, or T3 railguns, but that's about it. Alternatively, we could build dedicated carriers, but we've declined to build any battleship designs so far.

Long-term, we are embracing the Worm as it desires us, and progressing onto the path of psionic ascendancy, but both of those are many years away.

As you can see, our fleet capacity is far greater than what we've allocated to ships currently as we are keen not to attract the notice of one of the Fallen Empires by building too large of a military presence. As such, we're only at 344/561 for 39k fleet power. However, we've capped out our mineral stores at 28k and earn over 500 minerals a month, so we can grow our fleet quite quickly. We'd really, really, really like to avoid awakening either Fallen Empire, but especially the northern one, as the only real places for these empires to expand into is our territory. We're also mindful that conquering one empire could provoke the other one into action and that any campaign we initiate could be quite expensive. If we were to awaken one of the empires, or worse – trigger a War in Heaven, we'd expect the consequences to be devastating.

So our goal here is simple: conquer both Fallen Empires before they awaken.
How should we proceed to incorporate the worlds of these Fallen Empires into our domain so we may decontaminate these systems and protect them from other xenos that might show interest in them?