I know Latin fairly fluently and I was wondering the same thing since I'm not as familiar with this word. I believe capto, captare is the frequentative form of the verb capio, capere - to take. So if you are taking frequently, its like you're reaching for it, "trying to grab". A common example of a frequentative in english is commentate instead of comment. It kind of denotes a continuous action.
One I like in Latin is saltare - to dance, which comes from salire - to jump. Jumping continuosly is dancing!
Triclinium is a dining room in a domus (nice big family house). Cenaculum is an upper story or attic dining room, but it could also be thought of as a single family apartment room I believe. Most Romans lived in fairly small, cramped apartment buildings called "insula" - literally island. Which had several stories and rooms. Or if they had a shop (taberna), it would be on the first floor and their living/dining room would be upstairs
in Latin, every noun and adjective has three properties: Case, Number, and Gender. Case gives the function of the noun in the sentence. Number is whether it is singular or plural. and gender is masculine, feminine, or neuter. Adjectives must match the noun they modify in all three properties. In this sentence, meo is ablative, singular and masculine, so it must be modifying the word cenaculo which is also ablative, singular and masculine, and cannot modify mures, which is accusative, plural and common (can be either masculine or feminine)
If they did, then it was probably an attempt to imply the "tries to" sense without actually using the verb "to try;" and then it would have been changed, because it's likely to be missed by most students.
In any case, the actual "to grab (something)" is (aliquid) capiō, capere, cēpī, captum, while this is (aliquid) captō, captāre, captāvī, captātum.
A tutor once pointed out to me that with Latin having a relatively small vocabulary, its words have to work very hard. The OLD gives 9 definitions for capto.
To try to touch or take hold of, grasp at.
To catch at, (try to) draw in (air, breath; also, water); to seek to catch (wind).
To make for (a place); to try to reach (with a missile). To seek (shade, coolness, and other conditions).
To try to find or obtain, seek out. To look for or seize (an opportunity).
To seek, aim at. To seek to arouse or produce (in others).
To go in for, affect, aspire after (an attitude, course of action, etc.).
To try to capture. To seek to entrap (by military action). To seek to catch by hunting, fishing, etc. To try to catch (lovers, etc,).
To try to catch out or get the better of in argument, etc.
To try to win over or captivate, entice. To court the favour of, in the hope of securing a legacy; to seek (a legacy) by this means.
"try/long/aim for, desire; entice; hunt legacy; try to catch/grasp/seize/reach;"
I think it does have that meaning. Maybe you mean a different whitaker though
Capto, captare is the frequentative form of capio, capere, so to take/grab often is to "try to catch"