Will Lithium batteries mean we have to exploit Co-Mn-Ni nodules from the seafloor?
No it is not inevitable that the production and continued used of lithium batteries will mean the exploitation of cobalt, manganese and nickel nodules from the sea floor.
Sufficient quantities of these metals are obtained from conventional land based mines.
Nickel
Two types of nickel mineralization are currently mined: sulphides and oxides (laterites).
Sulphide nickel is easier to treat metallurgically than oxide nickels. Once mined nickel sulphides are initially processed using flotation plants. Nickel oxides usually need to be treated in autoclaves (high pressure cookers for minerals) and depending on secondary mineralization the autoclaves use either acid or ammonia.
The top ten nickel producing countries are:

The major laterite producers are: Indonesia, Philippines, New Caledonia, Brazil and Cuba.
The major sulphide producers are: Canada, Australia &amp; Russia.
Australia also has significant reserves of nickel laterites (oxides).
The top nickel mining companies are:

There is plenty of land based nickel available before sea bed nodules need to be mined for nickel.
Manganese
Manganese is the 12th most abundant element in the Earth’s crust. About 18.5 million tonnes of manganese are produced globally annually, from
South Africa   6.2 Mt
Australia      3.0 Mt
China          2.9 Mt
Gabon          1.8 Mt
Brazil         1.0 Mt

There are significant known resources of land based manganese for many years to come.
Cobalt
Cobalt is currently mined by eight countries, with the Congo being the main producer:
Congo           100.0 kt
Russia            6.1 kt
Australia         5.1 kt
Philippines       4.6 kt
Cuba              3.5 kt
Madagascar        3.3 kt
Papua New Guinea  3.1 kt
Canada            3.0 kt

Again, there is sufficient production and resources of land based Cobalt to not require the exploitation of sea bed nodules.