A PSS fare indicates the weekly 'base rate' applicable to a season ticket with a duration between one month and a year (bearing in mind, for instance, that a monthly costs 3.84× the weekly base rate, and an annual 40×).
Of those flows which have season tickets defined:
- Some have only a weekly season ticket fare (7DS) - and so this is what is used as the base rate for season tickets longer than a week. This is probably the most common arrangement, generally across shorter commutes or those not to/from/via London;
- Some have only a longer-period season ticket fare (PSS) - and so this means that weekly season tickets are not available. This is common where it is a particularly long distance journey, where the base rate is near to the cost of an Anytime Return (e.g. Manchester to London);
- Some have both a weekly and a longer-period season ticket fare (7DS and PSS) - and in most instances the PSS is anything from a few percent to a significant percentage less than the 7DS. This is most commonly found as the arrangement on some medium distance commutes, such as you have discovered here (or for example between the West Midlands and London).
Coming back to your question, it is not possible to buy a weekly season ticket at the PSS rate (unless they happen to be the same amount), so no, you cannot buy a weekly season ticket for £353.30.
Depending on the number of days you will be travelling, and whether or not you have any Railcards (or are eligible for any), it may be (significantly) cheaper to buy walk-up tickets for your travels - specifically, assuming you are travelling at times where "ordinary" Super Off-Peak and Off-Peak tickets to London are not valid, there are a number of tickets that
are valid at any time into/out of Paddington, despite being Off-Peak of one kind or another, and despite being cheaper than Anytime tickets to London.