I would say “steup” but it’s casual.
Otherwise you have to say “s’il vous plait”/”s’il te plait” !

It’s OK to write

S’il vous plaiiiiiiiiiiiiiiit

But “S’il vous plaiiiiiiiiiiiiiiit le faire ?” doesn’t make sense.
You could say:

Faites-le. S’il vous plaiiiiiiiiiiiiiiit ?
Do it. Pleeeeeeease ?

Is there a better way in French of conveying the English sense of Pleeeeeeeeeeeease at the beginning of a sentence?

Your question seems to contradict the title: you’re not asking for a more compact way, you’re asking for a more emphatic way; for example,

Faites-le, je vous en prie.
Faites-le donc, je vous en supplie.

As well as more emphatic, there are more formal ways,

Cher monsieur, aurez-vous SVP la gentillesse de le faire pour moi ?

The only example I can think of off-hand, of where I’d lengthen a vowel, is when literally shouting (calling) for help in an emergency:

• He-e-e-e-e-e-lp!
• Au secou-ou-ou-ou-ou-ou-ou-ou-ou-r!

One way to get the added emphasis/urgency in French would be by resorting to the different ways of “begging” someone to do something. For example:

• Je t’en prie (polite begging with the notion of “please” included)

• Je t’en supplie (a bit more urgent, but the “please” is still probably understood)

• Je t’implore (serious begging and the “please” is probably not understood)

• Je te pleure (serious begging, no “please” at all)

Or to stay with a slightly more concise variation of the “please” construction (granted it’s LESS, not more concise when repeated), you could get close to the idea of “pretty please” by repeating “S’te plait”:

“S’te plaît, s’te plaît !”, like a child begging his/her Mom for candy:

Maman, est-ce que je peux avoir deux bonbons ? S’te plaît, s’te plaît !

One more:

Siouplait. It sounds informal and quirky, as in replacing ‘understood’ by ‘okey-dokey’.

SVP doesn’t really sound long-winded to me, it’s ‘if-you-please’ (or more exactly ‘if it pleases you’). If you want to sound polite while not using it, you can use the conditional, same as in English ‘Could you give me the salt ?’ ‘Pourriez-vous me passer le sel ?’. With informal (and slightly incorrect English) ‘Can you give me the salt ?’ ‘Tu peux me passer le sel ?’. Or pedantic-formal ‘Would you be as kind, my good Sir, as to give me the salt ?’ ‘Auriez-vous l’obligeance, Monsieur, de me passer le sel ?’.

A clever way is to use the imperative form of the verb vouloir followed by the infinitive.

• Veuillez fermer la porte

• Veuillez ne pas fumer à bord du traversier

Or you can use the verb pouvoir indicative.

• Peux-tu faire ceci?

• Pouvez-vous faire ceci?

Or with conditional. (More polite)

• Pourriez-vous faire ceci?