You didn't say how much of the planet the Eel-Men were using, so I'll assume that they have densely populated cities and large uninhabited areas. There are lots of economics reasons why this would be expected. So, the first order of business is to figure out what needs to be destroyed, and what can/should be preserved. I presume the Eel-Men do not have enough resources to effectively patrol every cubic km of their ocean, and that defenses are concentrated around valuable targets. The interesting question is whether the Eel-Men leadership would put anti-orbital weapons on land to deny the use of land to the Lizard-Men, or if they would cede the land to concentrate on their precious underwater holdings.
Land-based defenses would be susceptible to orbital bombardment, so I assume that the Eel-Men would not bother. In this case, the Lizard-Men should try to establish a beach-head literally on the beach, but approached from space, rather than the water, because in this world, the sea is more dangerous than the air. Of course, this already presents a huge problem for the Eel-Men, because logistics is now much, much easier for the Lizard Marines. Instead of dropping a huge infantry force and suffering massive casualties, they can just drop AA batteries over the centers of continents, which should be the least defended, and transport them to the edges of the continents, where they can interdict any missiles launched from underwater. This should make it much more effective to drop troops and supplies. In this manner, they can achieve air/space superiority over continents with a fairly low casualty rate.
The Lizard-Men can now use their orbital/air superiority to eliminate Eel-Man observation stations above water. Now the Eel-Men don't even know where the Lizard-Men are massing to attack and lack early warning until they approach the water. Of course, the Lizard-Men would use props and disinformation tactics to make it look like they were gathering near one juicy target while actually massing towards another one. This would force the Eel-Men to spread their forces thin to defend all vulnerable cities, or to commit a strong defense to the most likely target(s). The Lizard-Men get to shape the battle space to their advantage.
The one benefit the Eel-Men do have is that they can lightly defend the cities furthest towards the center of the ocean, and gather forces mostly around the continents (you didn't mention whether the Eel-Men liked all depths, or preferred to live on the continental shelves).
Finally, the Lizard-Men attack one or more high value targets (ideally the ones which would most demoralize the defending army to force an early surrender). They would use a diversionary force to draw Eel-Men to the wrong location, and then strike the actual targets if the Eel-Men take the bait.
The real risk here is that once the Lizard-Men go underwater, they are at an immediate disadvantage. Their entire war machine is likely optimized for land warfare, since that is how they evolved. They would need to spend years training underwater before launching the invasion strike so that they will be prepared. Even so, the further underwater they go, the more risk they assume. Operating at 30m depth would be difficult and risky, but usually survivable. If the Eel-Men can dive to, say, 1000m, and that is where their cities are, then the Lizard-Men must always operate in submarines once they attack, and the Eel-Men only need to compromise their hulls to defeat them.
Obviously, torpedoes will be the most effective stand-off weapon, and the Lizard-Men would be well-advised to employ them liberally. Fielding big ships themselves would be needlessly expensive for the Eel-Men, as their best tactic would be swarming. Torpedoes are effective against large, slow-moving ships, but pretty useless against a swarm of 100 nearly-naked Eel-Men swimming at high speed and armed with attachable mines.
A torpedo requires a fairly large boat to carry it, so Lizard-Men will need decent-size submarines to bring the heavy firepower to the Eel-Man cities. Essentially, torpedoes are the only effective artillery the Lizard-Men can bring to bear on their underwater enemies, so large submarines are unavoidable. The threat of swarming swimmers means that they will also need point defense, which could be harpoon guns on turrets. However, this will become very expensive very quickly. Even if the harpoons are very small, the subs will not be able to carry an indefinite supply of them, and the Eel-Men can approach from literally every direction. Trying to defend an entire sphere of space at range is extremely difficult. If the effective range of the harpoons/fletchettes is only a dozen meters or so, then they will need many turrets all over the sub just to deal with swarmers. Of course, they can also eject depth charges to deal with un-armored swimmers, but again, there are only so many they can carry.
The Eel-Men can also bring ships to the fight, and they can quite effectively force the Lizard-men to fire their counter-measures with mass torpedoes. In fact, they don't even need to all be live. They can use mostly fake torpedoes to force defensive action, because the Lizard-Men will not be able to tell which are real and which are duds. Thus, the Eel-Men can clear the way for the swimmers, which then just swim up to the subs and plant charges, and swim away. They will take casualties, for sure, but they can make the fight very expensive for the Lizard-Men, who have to build the subs from scratch on land, while the Eel-Men have been building war machines for decades.
In fact, the Eel-Men may simply have leviathan ships which are massive torpedo boats which can overwhelm an entire squadron of Lizard subs' countermeasures. Surely the Eel-Men will have fought each other over sea floor and natural resources, so they should have this technology already. The ocean is a very inhospitable terrain for the Lizard-Men.
Although depth charges are quite damaging to unprotected swimmers, they are just as damaging if not more so to a sub operating near its maximum depth. If the Lizard-Men are required to dive 1000, 2000, 3000m to take key Eel-Man cities, then surely their subs will be vulnerable to even the slightest local overpressure caused by a detonating torpedo. This is like the effect where you can stand on an empty aluminum can, but if you just tap the side very lightly, it instantly collapses. Eel-Man torpedo barrages don't even need to hit the Lizard-Man subs. They just need to create enough shock wave near the subs to compromise their hull integrity. The weight of a 2000m water column will squish the subs like tin cans. This means that Lizard-Men will have to interdict all torpedoes at stand-off range, which seems nearly impossible. The lethal effective radius for an Eel-Man torpedo might be 100m or more, just because of overpressure deformation. Trying to keep all torpedoes outside a 100m sphere of a Lizard-Man sub might just be impossible at their level of technology.
I think the only ways the Lizard-Men can triumph are:
- The Eel-Men cannot naturally survive extreme depths and live on the continental shelf at no more than 100-200m depth.
- The Eel-Men have poor metallurgy skills and few or primitive war machines due to developing underwater (but this would make them less attractive for subjugation).
- The Lizard-Men have powerful industry and can drop raw metal on the planet from nearby asteroids and refine it quickly into war machines. Otherwise they have to mine it locally, which would be quite tedious.
- The Lizard-Men have more powerful computers, and can use acoustic counter-measures.
The last point is interesting, so I'll expand it a bit. Visual contact at depth will be difficult to impossible. If the Eel-Men live at depth, then they are probably capable of echolocation biologically, and may also be able to sense bioelectric fields for hunting prey. Torpedoes may also use sonar to locate their targets. On earth, subs try to hide, because there are few of them and the ocean is large. The Lizard-Men don't need to hide, because they are an invading army. They just need to disrupt enemy torpedoes. Thus, they can deploy acoustic drones which emit confounding sonar signals to muddy the battle-space. If Eel torpedoes can also get visual lock, then the Lizard subs and drones can also deploy ink counter-measures.
If Eel swimmers are acoustically sensitive due to their echolocation abilities, then Lizard-Man subs can just use intense sound as point defense against swimmers. The loud noise doesn't bother the Lizard-Men, but may actually be lethal to the delicate Eel-Men. And if they are bioelectric-sensitive, they could also use electrical shock near the sub to further repel Eel swimmers.
So, depending on the capabilities and technology level of each race, there are various weapons systems which can be deployed offensively and defensively. Choose their properties to decide what outcome you want!