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Season tickets from Eastleigh (ESL) to London Terminals are routed 'Any Permitted'. Therefore you can use all operators along all of the permitted routes from ESL to London - which include travelling via Reading (RDG) into London Paddington (PAD). In other words, travel via RDG would be permitted even if it weren't a strike day.
The only slight issue is that your season ticket is from ESL, whilst Southampton Airport Parkway (SOA) is one station south of there. However, one of the ways that a route can be permitted is for it to be within 3 miles of the shortest route; it's just under 1.5 miles from ESL to SOA, so 'doubling back' from ESL to SOA is within 3 miles of the shortest route to London. Accordingly I would argue it is a permitted route and you could travel that way even if it weren't a strike day.
That being said, it is possible that staff onboard the XC service will claim otherwise and will attempt to charge you for a new ticket. They may even try and charge you as far as Winchester or Basingstoke, wherever the next stop is. If you buy an extra ticket before boarding, you would only have to buy one as far as ESL because you can conduct a 'non-stop split' when one of your tickets is a season ticket (the idea is to allow you to extend your journey without being restricted in your choice of trains). As a SOA-ESL ticket is only a few pounds, you may decide to buy this to mitigate against any risk of being incorrectly charged.
In terms of the additional rights that you have on a strike day given that SWR likely won't be running any trains to or from Eastleigh, you are entitled to require them to re-route you at the earliest opportunity under Article 16 of the Passenger Rights and Obligations Regulation (PRO) - in other words, for them to make arrangements for your ticket to be accepted on other operators or routes to allow you to make your journey. Most train companies haven't given their staff any training on the PRO, so it's highly likely that this request will be wrongly denied unless it happens to align with ticket acceptance or other policy that's already been agreed. But if they refuse your re-routing request, you'd be entitled to recover any reasonable costs you incur as a result - for example, the above cost of a SOA-ESL ticket.
As you would be stranded without SWR's assistance, condition 28.2 of the National Rail Conditions of Travel (NRCoT) is also engaged, requiring any train company to help you complete your journey if they can reasonably do so. As XC will still be running services normally, they fall into this category and would again be obliged to accept your ticket to allow you to make your journey. So even if SWR refuse to re-route you under the PRO, and even if XC wrongly claim that your season isn't valid via SOA, they would still have to accept your ticket and allow you to travel for free under condition 28.2. Again, this is something that train companies don't tend to train their staff on properly so it's possible that an unsympathetic member of staff will wrongly attempt to charge you. But as above, this is something you'd be entitled to claim back if that did happen.
In summary, you should first contact SWR to ask them to re-route you under Article 16 of the PRO, which effectively entails arranging (or confirming) ticket acceptance with XC to allow you to travel from SOA. If they refuse to do this, you can choose whether you either board the XC service anyway (as it's within 3 miles of the shortest route so your season is valid, and they are obliged to assist you under condition 28.2 of the NRCoT), or you could just buy an additional SOA-ESL ticket and contact SWR or XC to claim this back afterwards.
In no scenario should you be left out of pocket, although I appreciate that if you do want to buy the SOA-ESL ticket for peace of mind, you may decide it's not worth your while to try and claim this small amount back.
