[SalesForce] Can you run a Batchable Apex bulk update as a different user

I would certainly regard it as an anti-pattern yes, the Salesforce documentation here, states this...

Use extreme care if you are planning to invoke a batch job from a trigger. You must be able to guarantee that the trigger will not add more batch jobs than the five that are allowed. In particular, consider API bulk updates, import wizards, mass record changes through the user interface, and all cases where more than one record can be updated at a time.

I personally think this statement does not go far enough to discourage this, as it basically breaks bulkificaiton, which as you know is a big no no. Unless your users of the given object fully accept and understand the implicatons (and its often hard to be confident they do tbh), i would never use it.

Instead, as you suggest create a scheduled job and have the trigger insert into some kind of work action object to process work in the background, then either chatter post or email notification of any failed work items back to the user, with a means to re-queue the work once they have resolved the issue.

There is of course on the horizon the most welcome Batch Apex FlexQueue. Note however this has its own limit of up to 100 queued items.

If the Apex flex queue has the maximum number (100) of jobs, this method returns an error and doesn’t place the job in the queue.

So even with this in place, my standard advice is to avoid linking Batch Jobs or @future work to granular user or API actions as you can rapidly flood the platform, resulting in failure to enqueue or the platform slowing overall processing down to compensate. Instead consider a more robust way to group such work with a solid user notification and error recovery solution in place.

If I were you I would just add the Ids in an after insert event.

if (trigger.isBefore)
{
    if (trigger.isInsert)
    {
        // do stuff once
    }
    if (trigger.isUpdate)
    {
        // do stuff once
        // track records already touched in this transaction
    }
}
if (trigger.isAfter)
{
    if (trigger.isInsert)
    {
        // track records already touched in this transaction
    }
}