Will BCH have problems with Ordinals or other "spam"?

No.

The only objective measurement of how "valid" a transaction is the fee that is paid by the transactor. Bitcoin Cash has low fees & anyone is welcome to transact as much as they want & for whatever reason they want provided they pay the (low) fee for each transaction.

If such activity is not economically valuable, the sender would not be willing to pay the required fee and/or will soon run out of money paying for unvaluable transactions at large volume. If they are able to continue this activity "in perpetuity" then clearly there must be some valuable economic activity being generated by it - even if other observers do not understand what it is. Such activity does not disrupt the network in the slightest or inconvenience any other users.

Bitcoin Cash is designed to reach global scale, nobody is able to afford to fill its enormous capacity in perpetuity.
If insufficient blockspace is available, ABLA will create more. If ABLA is a little slow, then network fees will rise marginally (say from $0.001 to $0.01), which is of virtually no concern to regular network users (who make only a few transactions per day) but is a 10x increase to a ""spammer"" operating at large volumes. Thus the network drains the funding of unneeded activity far faster than of more valuable transactions.
There will be no permanent problems with fees rising in any case.
For transactors that want to make & speculate on tokens, Bitcoin Cash has provided CashTokens, which are ultra-efficient & designed specifically for people interested in that use case to participate in the BCH network as much as they like!
Unlike Bitcoin BTC, there is no Inscriptions on BCH.
Unlike Bitcoin BTC, there is no fun to be had in trolling laser eye toxic maximalists who get upset about so-called "spam". The BCH community actively embraces & welcomes anyone to transact as much as they see fit, which ironically reduces the incentive for anyone to ""deliberately spam"" the network.

BTC Propaganda

Some converts from BTC struggle with the concept that allowing network participants to freely transact on the network wouldn't cause any issues. This is mostly a result of propaganda which has given them the incorrect mindset that blockspace is scarce & must be "used efficiently".

::: info

The foolishness of this "blockspace must be enforced in scarcity to preserve its value! This will help people to be conservative & not wasteful with it!" is easy to demonstrate as the opposite of good economics.

Imagine the government of your country decreed "Half of all gasoline is being confiscated & burned to ensure people are conservative with their usage of cars!".

This would of course skyrocket the price of gasoline & cause people to be more conservative with trips in their cars... crashing the economy & destroying everyone's standard of living totally unnecessarily.

The same logic applies to "We must preserve the blockspace!" No, a well-functioning economy provides the best service at the lowest price possible to satisfy the largest number of consumers it can.

:::

This is quite ironic, given that the BTC community have (inadvertently) done everything they can to make BTC LESS efficient.

Restricted blocksize makes the market for on-chain fees high & in constant flux, leaving network participants to operate in an inefficient disorganised mess.
Replace By Fee encourages even further chaos by allowing inefficient resubmissions of transactions in a constant war to knock other transactions out of the mempool.
The SegWit ""upgrade"" discounted the storing of arbitrary data in the witness area of the BTC protocol, giving data storage an unnatural cost advantage over "financial transactions".
The Taproot ""upgrade"" allowed arbitrary data to be stored in the witness area of the BTC protocol as Inscriptions.

None of these problems apply to BCH. It does not have Restricted Blocksize, RBF, SegWit or Taproot. Bitcoin was actually designed to be ultra efficient & at scale the combined transaction fees will perfectly match commodity costs of sending & storing those transactions across the redundancy of the node network. On an efficient Bitcoin designed for transactions & without all these damaging ""upgrades"", people making transactions is of course not a problem, it is the entire point & highly encouraged!

BTC Propaganda​

BTC Propaganda