Though not the answer, I’d like to put some light on whether nodes should or should not be rewarded for validation:
using a light node wallet puts the wallet itself at (limited) risk rather than the whole network. For example even if all full nodes are malicious, users can resort to running their own full-nodes (there are less likely fork scenarios though). Users are responsible for security of their light wallets in the same way they responsible for security of their private keys.
The network can survive and keep being decentralized (and contain valid data) as long as there is no monopoly on mining and at least one full copy of block-chain(s) exists (miners or at least mining pools usually have full copy).
SPV (simple payment verification) is robust to attacks aimed at stealing funds directly from wallet if you follow n-confirmation rule. Light wallets are less robust to eclipse attacks, 1-confirmation attacks (full node can hide new transactions and longest chains from users) and maybe some more exotic types.
there are non-monetary incentives: https://en.bitcoin.it/wiki/Clearing_Up_Misconceptions_About_Full_Nodes#Myth:_There_is_no_incentive_to_run_nodes_so_the_network_relies_on_altruism
P. S. with that said while I’m not a big fan of PoS - using simplified overlay version of PoS (operator/stakeholder locks funds conditioned on its own “good behavior”) combined with direct rewards as a secondary security measure to increase availability, prevent sybil-attacks and de-incentivize full nodes from cheating could work and stimulate running more full nodes.
Although as it mentioned in other answers - proving that the node is full is hard to impossible, so it would only guarantee that one node doesn’t pretend to be a group of nodes and it could also give mechanisms for penalizing malicious nodes (prevent them from censorship for example). The node would be rewarded exclusively for broadcasting transactions (and penalized for not doing so), they would not be rewarded for other light-node to full-node interactions (like requesting a history etc).
Such hypothetical mechanism would not require changes to bitcoin consensus or existing protocols, it would only require change of light wallet’s signing logic (the users wallet would add additional “full node reward output” in transaction) and maybe some overlay protocols to penalize censorship and unavailability (it would have to be overlay because bitcoin Script is not practically powerful enough to check proofs inside contracts).
However, again as it was mentioned, merchants and wallet providers are not monopolized and naturally abundant (thanks to competition) so introducing additional full-node fees for this type of security may be a little bit too much, but who knows :).
Note: In theory, PoS as secondary verifier would also improve safety of 1-confirmation transactions as was argued in earliest (before Eth and others were even conceived) btc-related discussions involving @QuantumMechanic, but in practice stakeholder’s “approval” is incompatible with PoW (stakeholder/SPO has no way of knowing if miners would continue building on top of block stakeholder/SPO chose). So it could only be applicable to prevent Sybil attacks, penalize censorship and increase availability.