There are two ways to deal with this problem:
1) Force every user to submit PoW with their transactions, i.e. no transaction gets on the block chain without PoW attached. Note getting this sort of design to be robust, requires an entirely different way of structuring a block chain. If the attached PoW is low enough difficulty, then it costs more to farm it out (network latency cost) than to mine it locally given it is an insignificant and unnoticeable cost.
2) Limit debasement to a small annual percentage.
In that case, the professional miner will not be able to mine a significant quantity of the coins, and they will not be selling a significant percentage of the market cap. Thus the downward pressure on the price that impacts Bitcoin will be abated.
The reason for the block reward is to subsidise the security of the blockchain. In bitcoin, each transaction would need to pay $7 of transaction fees to achieve the level of security that it currently enjoys, without block reward.
Point 1 - if only transaction submitters can mine their own blocks, how do you handle difficulty adjustment?
Difficulty is adjusted as it is always is for PoW.
Point 2 - If it is not profitable to mine the chain, how do you achieve the same level of security as with a subsidised chain?
It might still be profitable to mine the chain for the professionals with very low electricity costs and very efficient ASICs, but for the transaction submitters it surely isn't profitable (but it isn't noticeable either so they don't care or even know).
The profitability is orthogonal to your point, which is you mean how to get Bitcoin's security if the percentage of your market cap paid to mining debasement and the market cap are not the same level as for Bitcoin.
You are making the point that small market cap and/or low debasement coins have lower PoW security.
I already asserted that 51% attacks are pretty much impotent. It will require nearer to 100% attack to snuff out the minority and force them to adopt a protocol change, which is the major risk from a 51% attack in existing PoW designs. Also in my design 51% doesn't help for creating a double-spend.
The only need for the PoW in my design is to prevent a Sybil attack on the distributed confirmation resources. A 51% attack that orphans a legitimate chain of these statements about resources, can't undo the reality of the inertia that has been established on that orphaned chain. It can supplement the resources, but attempting to take away resources that already intertwined in the inertia will be ignored by all those nodes which are bound to lose income from unwinding that inertia. In other words, there will be sufficient allowance made for any reasonable orphan rate due to normal propagation delays and anything outside of that will have already accrued inertia and be impossible for the 51% attack to unwind in order to create a double-spend.
The reason that inertia alone couldn't be used to establish the single-point-of-truth on the resource set is because it doesn't have a definable boundary such as blocks that can be counted. Inertia is a local perspective (each participating node has a perspective on the inertia that is invested in) but it doesn't have global consistency. This is why I am confident the DAG coins such as Iota are flawed. Thus I still needed PoW for this global consensus, but the 51% can't win every block (only near to 100% can), so it is impotent in my design. The 51% can blacklist the minority's block solutions, but if the reasonable propagation delay is orders-of-magnitude less than the period of 1 block then such habitual blacklisting can be statistically distinguished from orphan rate and thus can be objectively identified as malicious and ignored. Propagation ends up being the crucial design factor in design like mine. This is why I said I don't think Bitcoin can graft these things onto their existing paradigm. More details will forthcoming in the white paper.
That is far more details than I really wanted to release now. So any further questions that require explaining the details of my design might illicit from me, "wait for the white paper".
Edit: Bitcoin pays far too much to security. The only reason to pay anything for security in my design is because 0% debasement is deflationary because users lose private keys. By paying those losses back to users who transact, it transfers value over time from those who don't transact to those who do, which is favorable for encouraging more currency use and more network effects. What could be done instead is pay nothing for PoW mining and then pay debasement proportionally to every coin that transacted in that period (or pay weighted by coin days destroyed age). But this weighting by value would be incongruent with hidden values (private data). Instead the payment could be weighted by the number of transactions, which is functionally equivalent to paying for each mining share of PoW (assuming every transaction is required to include the same level of PoW). If the PoW allowed to be submitted with each transaction is less than the profitability for the confirmation node from typical transaction fee, then professional miners could not mine profitably. They might consider being their own confirmation node, except the assumption is transaction fees will in a competitive environment be very near to cost with very low profit margins so if the PoW was sufficiently high then professional miner couldn't overcome. But I think that is unlikely to be the case because of the asymmetry in the delay for a home user and cost for a professional miner to compute that PoW share. If private keys timed out (e.g. yearly), we could more precisely calculate the level of debasement needed, but this would be shocking to people that lose their coins because they were inactive. Instead if coins assumed to be lost were allowed to be spent, one could use demurrage to recapture excess prior debasement, but the problem is that doesn't spread the pain out equitably between those who were formerly invested in the coin but sold and those who are currently invested. Debasement is much less individually and immediately noticeable. Small levels of debasement are not an issue for users (nor investors). Heck Bitcoin's debasement is still nearly 10% per year and was much higher in the past. I think many people forget that many coins are lost (I've lost private keys for close to 1 BTC in 2 years already which is roughly 1+% of the volume of BTC that ever passed through my hands). With smaller balances and microtransactions, much larger percentage will be lost.