First of all, this is not a very good piece of music. It's just a dry and boring transcription of the main theme played at a slow tempo. It's not worth spending more energy worrying about technique than the arranger spent writing it.
That being said, many people are overly obsessed with the sostenuto (middle) pedal, and with doing weird hand gymnastics to dutifully keep fingers on the keys for the full written durations. The real solution is to just use the damper pedal. Keep everything in the hands indicated. If you have to clear the pedal before the end of the bar to keep it from getting muddy, that's fine. This is also where half-pedaling can be helpful.
Here's where everyone downvotes me and tells me I'm wrong in the comments.
"But what about the staccatos????"
(1) You're putting more thought into this than the arranger did. (2) You can create the illusion of staccato notes even with the damper pedal down, and again half-pedaling helps with this.
"But you have to sustain the whole notes for the whole bar!"
No you don't. That's just not how piano music works. Piano notes start decaying immediately, and our brains do a lot to create the illusion of sustain when it's not actually there. It's a normal notational shorthand to just slap a whole note on a measure to indicate "this is the chord for the measure"; you're not disgracing the piece if you don't manage to keep it ringing the whole time.