你的怀疑是有道理的，就这个句子而言，把不定式看作宾语补语或定语都可以。

但在其他情况下，with复合结构中的宾语补语不宜分析为定语。如：

She sleeps with the light on. 她爱开着灯睡觉。

I was dancing with both children in my arms. 我抱着两个孩子跳舞。

He was asleep with his head on his arms. 他头枕着手臂睡着了。

It’s not polite to talk with your mouth full. 满嘴食物时讲话是不礼貌的。

with 的复合结构或者独立主格结构后面不是作定语而是作宾补成分。

with +名词+(不定式to do/副词/形容词/过去分词/过去分词/介词短语）

(1) 不定式

With a lot of work to do, Tom can't go out to play.

I can't go out with some clothes to wash.

I had to go to bed with nothing to do.

With the boy to lead the way,we will find the house easily tomorrow.

(2) 副词

Don't leave the room with lights on.

With the meal over, we all went home.

His mother sat in an armchair with her head down.

(3)形容词

He left the house with the windows open.

She came into the room, with her nose red.

She lay in bed with her face pale.

(4)过去分词

He is used to sleeping with the window closed.

He lay on the bed with the bedroom door shut.

With everything done, she went home.

With his key lost, the couldn't get into the office.

(5)现在分词

They start working with the machine running.

She fell asleep with the light burning.

With the machine helping us, we could finish the work on time

With the boy leading the way,we found he house easily.

(6)介词短语

The teacher came into the classroom with a book in his hand.

The master was walking up and down with the ruler under his arm。

She left the office with tears in her eyes.

特备备注：以上都属于宾补结构。

(7)名词

He died with his daughter yet a school girl.