Did the Scottish king deliberately send Sir Patrick to his doom?

While there is not enough detail to settle the question fully, it does not seem likely that the King of Scotland deliberately intended harm to Sir Patrick Spens. Instead, we see in this poem the familiar problem of too great a distance between those who command and those who obey, the fatal separation between the generals who give orders and the privates who must carry them out. The king has an urgent mission that he wants completed, involving the movement of "the king's daugher of Noroway" to Scotland:

THE king sits in Dunfermline town
Drinking the blude-red wine;
'O whare will I get a skeely skipper
To sail this new ship o' mine?'

Note the way the king has asked the question: he has demanded to be told of a "skeely" (skilful) captain. Given this, we need not doubt that the advice of the "old knight" is sincere:

'Sir Patrick Spens is the best sailor
That ever sail'd the sea.'

The asked question has been answered, and a skilful sailor nominated. The question that no one has asked, no doubt because it has occured to no one, is whether it is a good idea to go on a voyage at all at this time of year.

When Sir Patrick receives the order, we note that he laments it loudly, but does not think of questioning it:

'O wha is this has done this deed
And tauld the king o' me,
To send us out, at this time o' year,
To sail upon the sea?

It is of course possible that the King has meant Sir Patrick's doom, or that the person who nominated him did it with malicioius intent, but on the other hand, the King has sent Sir Patrick on a royal mission whose failure will reflect badly on the King himself, will rob him of a number of his lords, and will probably strain relations with Norway, which will not appreciate its princess being drowned due to Scottish carelessness. Again, the person who makes the nomination is an old knight, not a sailor. It seems more likely that the King and his old knight have concentrated too narrowly on the mere question of skill, and have forgotton or disregarded how difficult it must be for a man like Sir Patrick to question or defy a direct royal order delivered in writing and presented as a matter of urgency.